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Theorem br1steqgOLD 32120
Description: Obsolete version of br1steqg 7390 as of 9-Feb-2022. (Contributed by Scott Fenton, 2-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
br1steqgOLD ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))

Proof of Theorem br1steqgOLD
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 4561 . . . . . 6 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21breq1d 4821 . . . . 5 (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩1st 𝐶 ↔ ⟨𝐴, 𝑦⟩1st 𝐶))
3 eqeq2 2776 . . . . 5 (𝑥 = 𝐴 → (𝐶 = 𝑥𝐶 = 𝐴))
42, 3bibi12d 336 . . . 4 (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩1st 𝐶𝐶 = 𝑥) ↔ (⟨𝐴, 𝑦⟩1st 𝐶𝐶 = 𝐴)))
54imbi2d 331 . . 3 (𝑥 = 𝐴 → ((𝐶𝑋 → (⟨𝑥, 𝑦⟩1st 𝐶𝐶 = 𝑥)) ↔ (𝐶𝑋 → (⟨𝐴, 𝑦⟩1st 𝐶𝐶 = 𝐴))))
6 opeq2 4562 . . . . . 6 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
76breq1d 4821 . . . . 5 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩1st 𝐶 ↔ ⟨𝐴, 𝐵⟩1st 𝐶))
87bibi1d 334 . . . 4 (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩1st 𝐶𝐶 = 𝐴) ↔ (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)))
98imbi2d 331 . . 3 (𝑦 = 𝐵 → ((𝐶𝑋 → (⟨𝐴, 𝑦⟩1st 𝐶𝐶 = 𝐴)) ↔ (𝐶𝑋 → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))))
10 breq2 4815 . . . 4 (𝑧 = 𝐶 → (⟨𝑥, 𝑦⟩1st 𝑧 ↔ ⟨𝑥, 𝑦⟩1st 𝐶))
11 eqeq1 2769 . . . 4 (𝑧 = 𝐶 → (𝑧 = 𝑥𝐶 = 𝑥))
12 vex 3353 . . . . 5 𝑥 ∈ V
13 vex 3353 . . . . 5 𝑦 ∈ V
1412, 13br1steq 32118 . . . 4 (⟨𝑥, 𝑦⟩1st 𝑧𝑧 = 𝑥)
1510, 11, 14vtoclbg 3419 . . 3 (𝐶𝑋 → (⟨𝑥, 𝑦⟩1st 𝐶𝐶 = 𝑥))
165, 9, 15vtocl2g 3422 . 2 ((𝐴𝑉𝐵𝑊) → (𝐶𝑋 → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)))
17163impia 1145 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  w3a 1107   = wceq 1652  wcel 2155  cop 4342   class class class wbr 4811  1st c1st 7366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fo 6076  df-fv 6078  df-1st 7368
This theorem is referenced by: (None)
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