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Theorem tratrbVD 41488
Description: Virtual deduction proof of tratrb 41163. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1:: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)    ▶   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) 𝐵𝐴)   )
2:1,?: e1a 41254 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   Tr 𝐴   )
3:1,?: e1a 41254 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)    ▶   𝑥𝐴𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
4:1,?: e1a 41254 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝐵𝐴   )
5:: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   (𝑥𝑦𝑦𝐵)   )
6:5,?: e2 41258 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑥𝑦   )
7:5,?: e2 41258 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑦𝐵   )
8:2,7,4,?: e121 41283 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑦𝐴   )
9:2,6,8,?: e122 41280 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑥𝐴   )
10:: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵), 𝐵𝑥   ▶   𝐵𝑥   )
11:6,7,10,?: e223 41262 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵), 𝐵𝑥   ▶   (𝑥𝑦𝑦𝐵𝐵𝑥)   )
12:11: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))   )
13:: ¬ (𝑥𝑦𝑦𝐵 𝐵𝑥)
14:12,13,?: e20 41354 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   ¬ 𝐵𝑥   )
15:: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵), 𝑥 = 𝐵   ▶   𝑥 = 𝐵   )
16:7,15,?: e23 41382 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵), 𝑥 = 𝐵   ▶   𝑦𝑥   )
17:6,16,?: e23 41382 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵), 𝑥 = 𝐵   ▶   (𝑥𝑦𝑦𝑥)   )
18:17: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   (𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥))   )
19:: ¬ (𝑥𝑦𝑦𝑥)
20:18,19,?: e20 41354 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   ¬ 𝑥 = 𝐵   )
21:3,?: e1a 41254 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑦𝐴 𝑥𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
22:21,9,4,?: e121 41283 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥 𝑥 = 𝑦)   )
23:22,?: e2 41258 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
24:4,23,?: e12 41351 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   (𝑥𝐵𝐵𝑥𝑥 = 𝐵)   )
25:14,20,24,?: e222 41263 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑥𝐵   )
26:25: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   ((𝑥𝑦 𝑦𝐵) → 𝑥𝐵)   )
27:: (∀𝑥𝐴𝑦𝐴(𝑥𝑦 𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝑥𝐴𝑦𝐴(𝑥𝑦 𝑦𝑥𝑥 = 𝑦))
28:27,?: e0a 41399 ((Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴(𝑥𝑦𝑦𝑥 𝑥 = 𝑦) ∧ 𝐵𝐴))
29:28,26: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)    ▶   𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
30:: (∀𝑥𝐴𝑦𝐴(𝑥𝑦 𝑦𝑥𝑥 = 𝑦) → ∀𝑥𝑥𝐴𝑦𝐴(𝑥𝑦 𝑦𝑥𝑥 = 𝑦))
31:30,?: e0a 41399 ((Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴))
32:31,29: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑥 𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
33:32,?: e1a 41254 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   Tr 𝐵   )
qed:33: ((Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tratrbVD ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem tratrbVD
StepHypRef Expression
1 hbra1 3215 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑥𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
2 alrim3con13v 41160 . . . . 5 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑥𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)))
31, 2e0a 41399 . . . 4 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴))
4 ax-5 1912 . . . . . . 7 (𝑥𝐴 → ∀𝑦 𝑥𝐴)
5 hbra1 3215 . . . . . . 7 (∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
64, 5hbral 3216 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
7 alrim3con13v 41160 . . . . . 6 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)))
86, 7e0a 41399 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴))
9 idn2 41240 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   (𝑥𝑦𝑦𝐵)   )
10 simpl 486 . . . . . . . . . . 11 ((𝑥𝑦𝑦𝐵) → 𝑥𝑦)
119, 10e2 41258 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑥𝑦   )
12 simpr 488 . . . . . . . . . . 11 ((𝑥𝑦𝑦𝐵) → 𝑦𝐵)
139, 12e2 41258 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑦𝐵   )
14 idn3 41242 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝐵𝑥   ▶   𝐵𝑥   )
15 pm3.2an3 1337 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝐵 → (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))))
1611, 13, 14, 15e223 41262 . . . . . . . . 9 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝐵𝑥   ▶   (𝑥𝑦𝑦𝐵𝐵𝑥)   )
1716in3 41236 . . . . . . . 8 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))   )
18 en3lp 9075 . . . . . . . 8 ¬ (𝑥𝑦𝑦𝐵𝐵𝑥)
19 con3 156 . . . . . . . 8 ((𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥)) → (¬ (𝑥𝑦𝑦𝐵𝐵𝑥) → ¬ 𝐵𝑥))
2017, 18, 19e20 41354 . . . . . . 7 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶    ¬ 𝐵𝑥   )
21 idn3 41242 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝑥 = 𝐵   ▶   𝑥 = 𝐵   )
22 eleq2 2904 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
2322biimprcd 253 . . . . . . . . . . 11 (𝑦𝐵 → (𝑥 = 𝐵𝑦𝑥))
2413, 21, 23e23 41382 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝑥 = 𝐵   ▶   𝑦𝑥   )
25 pm3.2 473 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝑥 → (𝑥𝑦𝑦𝑥)))
2611, 24, 25e23 41382 . . . . . . . . 9 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝑥 = 𝐵   ▶   (𝑥𝑦𝑦𝑥)   )
2726in3 41236 . . . . . . . 8 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   (𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥))   )
28 en2lp 9067 . . . . . . . 8 ¬ (𝑥𝑦𝑦𝑥)
29 con3 156 . . . . . . . 8 ((𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥)) → (¬ (𝑥𝑦𝑦𝑥) → ¬ 𝑥 = 𝐵))
3027, 28, 29e20 41354 . . . . . . 7 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶    ¬ 𝑥 = 𝐵   )
31 idn1 41201 . . . . . . . . 9 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   )
32 simp3 1135 . . . . . . . . 9 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → 𝐵𝐴)
3331, 32e1a 41254 . . . . . . . 8 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝐵𝐴   )
34 simp2 1134 . . . . . . . . . . . 12 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
3531, 34e1a 41254 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
36 ralcom 3346 . . . . . . . . . . . 12 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
3736biimpi 219 . . . . . . . . . . 11 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
3835, 37e1a 41254 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
39 simp1 1133 . . . . . . . . . . . 12 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐴)
4031, 39e1a 41254 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   Tr 𝐴   )
41 trel 5166 . . . . . . . . . . . . 13 (Tr 𝐴 → ((𝑦𝐵𝐵𝐴) → 𝑦𝐴))
4241expd 419 . . . . . . . . . . . 12 (Tr 𝐴 → (𝑦𝐵 → (𝐵𝐴𝑦𝐴)))
4340, 13, 33, 42e121 41283 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑦𝐴   )
44 trel 5166 . . . . . . . . . . . 12 (Tr 𝐴 → ((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
4544expd 419 . . . . . . . . . . 11 (Tr 𝐴 → (𝑥𝑦 → (𝑦𝐴𝑥𝐴)))
4640, 11, 43, 45e122 41280 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑥𝐴   )
47 rspsbc2 41161 . . . . . . . . . . 11 (𝐵𝐴 → (𝑥𝐴 → (∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))))
4847com13 88 . . . . . . . . . 10 (∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝑥𝐴 → (𝐵𝐴[𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))))
4938, 46, 33, 48e121 41283 . . . . . . . . 9 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
50 equid 2020 . . . . . . . . . . 11 𝑥 = 𝑥
51 sbceq2a 3771 . . . . . . . . . . 11 (𝑥 = 𝑥 → ([𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
5250, 51ax-mp 5 . . . . . . . . . 10 ([𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))
5352biimpi 219 . . . . . . . . 9 ([𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) → [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))
5449, 53e2 41258 . . . . . . . 8 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
55 sbcoreleleq 41162 . . . . . . . . 9 (𝐵𝐴 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
5655biimpd 232 . . . . . . . 8 (𝐵𝐴 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
5733, 54, 56e12 41351 . . . . . . 7 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   (𝑥𝐵𝐵𝑥𝑥 = 𝐵)   )
58 3ornot23 41136 . . . . . . . 8 ((¬ 𝐵𝑥 ∧ ¬ 𝑥 = 𝐵) → ((𝑥𝐵𝐵𝑥𝑥 = 𝐵) → 𝑥𝐵))
5958ex 416 . . . . . . 7 𝐵𝑥 → (¬ 𝑥 = 𝐵 → ((𝑥𝐵𝐵𝑥𝑥 = 𝐵) → 𝑥𝐵)))
6020, 30, 57, 59e222 41263 . . . . . 6 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑥𝐵   )
6160in2 41232 . . . . 5 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   ((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
628, 61gen11nv 41244 . . . 4 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
633, 62gen11nv 41244 . . 3 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
64 dftr2 5161 . . . 4 (Tr 𝐵 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
6564biimpri 231 . . 3 (∀𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵) → Tr 𝐵)
6663, 65e1a 41254 . 2 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   Tr 𝐵   )
6766in1 41198 1 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3o 1083  w3a 1084  wal 1536   = wceq 1538  wcel 2115  wral 3133  [wsbc 3759  Tr wtr 5159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7456  ax-reg 9054
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-tp 4556  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-tr 5160  df-eprel 5453  df-fr 5502  df-vd1 41197  df-vd2 41205  df-vd3 41217
This theorem is referenced by: (None)
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