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Theorem tratrbVD 39585
Description: Virtual deduction proof of tratrb 39238. 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 39344 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   Tr 𝐴   )
3:1,?: e1a 39344 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)    ▶   𝑥𝐴𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
4:1,?: e1a 39344 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝐵𝐴   )
5:: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   (𝑥𝑦𝑦𝐵)   )
6:5,?: e2 39348 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑥𝑦   )
7:5,?: e2 39348 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑦𝐵   )
8:2,7,4,?: e121 39373 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑦𝐴   )
9:2,6,8,?: e122 39370 (   (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 39352 (   (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 39445 (   (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 39473 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵), 𝑥 = 𝐵   ▶   𝑦𝑥   )
17:6,16,?: e23 39473 (   (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 39445 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   ¬ 𝑥 = 𝐵   )
21:3,?: e1a 39344 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑦𝐴 𝑥𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
22:21,9,4,?: e121 39373 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥 𝑥 = 𝑦)   )
23:22,?: e2 39348 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
24:4,23,?: e12 39442 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   (𝑥𝐵𝐵𝑥𝑥 = 𝐵)   )
25:14,20,24,?: e222 39353 (   (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 39490 ((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 39490 ((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 39344 (   (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 3130 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑥𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
2 alrim3con13v 39235 . . . . 5 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑥𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)))
31, 2e0a 39490 . . . 4 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴))
4 ax-5 2001 . . . . . . 7 (𝑥𝐴 → ∀𝑦 𝑥𝐴)
5 hbra1 3130 . . . . . . 7 (∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
64, 5hbral 3131 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
7 alrim3con13v 39235 . . . . . 6 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)))
86, 7e0a 39490 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴))
9 idn2 39330 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   (𝑥𝑦𝑦𝐵)   )
10 simpl 470 . . . . . . . . . . 11 ((𝑥𝑦𝑦𝐵) → 𝑥𝑦)
119, 10e2 39348 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑥𝑦   )
12 simpr 473 . . . . . . . . . . 11 ((𝑥𝑦𝑦𝐵) → 𝑦𝐵)
139, 12e2 39348 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑦𝐵   )
14 idn3 39332 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝐵𝑥   ▶   𝐵𝑥   )
15 pm3.2an3 1432 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝐵 → (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))))
1611, 13, 14, 15e223 39352 . . . . . . . . 9 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝐵𝑥   ▶   (𝑥𝑦𝑦𝐵𝐵𝑥)   )
1716in3 39326 . . . . . . . 8 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))   )
18 en3lp 8752 . . . . . . . 8 ¬ (𝑥𝑦𝑦𝐵𝐵𝑥)
19 con3 150 . . . . . . . 8 ((𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥)) → (¬ (𝑥𝑦𝑦𝐵𝐵𝑥) → ¬ 𝐵𝑥))
2017, 18, 19e20 39445 . . . . . . 7 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶    ¬ 𝐵𝑥   )
21 idn3 39332 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝑥 = 𝐵   ▶   𝑥 = 𝐵   )
22 eleq2 2874 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
2322biimprcd 241 . . . . . . . . . . 11 (𝑦𝐵 → (𝑥 = 𝐵𝑦𝑥))
2413, 21, 23e23 39473 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝑥 = 𝐵   ▶   𝑦𝑥   )
25 pm3.2 457 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝑥 → (𝑥𝑦𝑦𝑥)))
2611, 24, 25e23 39473 . . . . . . . . 9 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝑥 = 𝐵   ▶   (𝑥𝑦𝑦𝑥)   )
2726in3 39326 . . . . . . . 8 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   (𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥))   )
28 en2lp 8745 . . . . . . . 8 ¬ (𝑥𝑦𝑦𝑥)
29 con3 150 . . . . . . . 8 ((𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥)) → (¬ (𝑥𝑦𝑦𝑥) → ¬ 𝑥 = 𝐵))
3027, 28, 29e20 39445 . . . . . . 7 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶    ¬ 𝑥 = 𝐵   )
31 idn1 39282 . . . . . . . . 9 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   )
32 simp3 1161 . . . . . . . . 9 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → 𝐵𝐴)
3331, 32e1a 39344 . . . . . . . 8 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝐵𝐴   )
34 simp2 1160 . . . . . . . . . . . 12 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
3531, 34e1a 39344 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
36 ralcom2 3292 . . . . . . . . . . 11 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
3735, 36e1a 39344 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
38 simp1 1159 . . . . . . . . . . . 12 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐴)
3931, 38e1a 39344 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   Tr 𝐴   )
40 trel 4953 . . . . . . . . . . . . 13 (Tr 𝐴 → ((𝑦𝐵𝐵𝐴) → 𝑦𝐴))
4140expd 402 . . . . . . . . . . . 12 (Tr 𝐴 → (𝑦𝐵 → (𝐵𝐴𝑦𝐴)))
4239, 13, 33, 41e121 39373 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑦𝐴   )
43 trel 4953 . . . . . . . . . . . 12 (Tr 𝐴 → ((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
4443expd 402 . . . . . . . . . . 11 (Tr 𝐴 → (𝑥𝑦 → (𝑦𝐴𝑥𝐴)))
4539, 11, 42, 44e122 39370 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑥𝐴   )
46 rspsbc2 39236 . . . . . . . . . . 11 (𝐵𝐴 → (𝑥𝐴 → (∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))))
4746com13 88 . . . . . . . . . 10 (∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝑥𝐴 → (𝐵𝐴[𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))))
4837, 45, 33, 47e121 39373 . . . . . . . . 9 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
49 equid 2108 . . . . . . . . . . 11 𝑥 = 𝑥
50 sbceq2a 3645 . . . . . . . . . . 11 (𝑥 = 𝑥 → ([𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
5149, 50ax-mp 5 . . . . . . . . . 10 ([𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))
5251biimpi 207 . . . . . . . . 9 ([𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) → [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))
5348, 52e2 39348 . . . . . . . 8 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
54 sbcoreleleq 39237 . . . . . . . . 9 (𝐵𝐴 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
5554biimpd 220 . . . . . . . 8 (𝐵𝐴 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
5633, 53, 55e12 39442 . . . . . . 7 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   (𝑥𝐵𝐵𝑥𝑥 = 𝐵)   )
57 3ornot23 39207 . . . . . . . 8 ((¬ 𝐵𝑥 ∧ ¬ 𝑥 = 𝐵) → ((𝑥𝐵𝐵𝑥𝑥 = 𝐵) → 𝑥𝐵))
5857ex 399 . . . . . . 7 𝐵𝑥 → (¬ 𝑥 = 𝐵 → ((𝑥𝐵𝐵𝑥𝑥 = 𝐵) → 𝑥𝐵)))
5920, 30, 56, 58e222 39353 . . . . . 6 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑥𝐵   )
6059in2 39322 . . . . 5 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   ((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
618, 60gen11nv 39334 . . . 4 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
623, 61gen11nv 39334 . . 3 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
63 dftr2 4948 . . . 4 (Tr 𝐵 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
6463biimpri 219 . . 3 (∀𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵) → Tr 𝐵)
6562, 64e1a 39344 . 2 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   Tr 𝐵   )
6665in1 39279 1 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3o 1099  w3a 1100  wal 1635   = wceq 1637  wcel 2156  wral 3096  [wsbc 3633  Tr wtr 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096  ax-un 7175  ax-reg 8732
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-tr 4947  df-eprel 5224  df-fr 5270  df-vd1 39278  df-vd2 39286  df-vd3 39298
This theorem is referenced by: (None)
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