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Theorem br1steqgOLD 32010
 Description: Obsolete version of br1steqg 7337 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 4539 . . . . . 6 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21breq1d 4796 . . . . 5 (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩1st 𝐶 ↔ ⟨𝐴, 𝑦⟩1st 𝐶))
3 eqeq2 2782 . . . . 5 (𝑥 = 𝐴 → (𝐶 = 𝑥𝐶 = 𝐴))
42, 3bibi12d 334 . . . 4 (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩1st 𝐶𝐶 = 𝑥) ↔ (⟨𝐴, 𝑦⟩1st 𝐶𝐶 = 𝐴)))
54imbi2d 329 . . 3 (𝑥 = 𝐴 → ((𝐶𝑋 → (⟨𝑥, 𝑦⟩1st 𝐶𝐶 = 𝑥)) ↔ (𝐶𝑋 → (⟨𝐴, 𝑦⟩1st 𝐶𝐶 = 𝐴))))
6 opeq2 4540 . . . . . 6 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
76breq1d 4796 . . . . 5 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩1st 𝐶 ↔ ⟨𝐴, 𝐵⟩1st 𝐶))
87bibi1d 332 . . . 4 (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩1st 𝐶𝐶 = 𝐴) ↔ (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)))
98imbi2d 329 . . 3 (𝑦 = 𝐵 → ((𝐶𝑋 → (⟨𝐴, 𝑦⟩1st 𝐶𝐶 = 𝐴)) ↔ (𝐶𝑋 → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))))
10 breq2 4790 . . . 4 (𝑧 = 𝐶 → (⟨𝑥, 𝑦⟩1st 𝑧 ↔ ⟨𝑥, 𝑦⟩1st 𝐶))
11 eqeq1 2775 . . . 4 (𝑧 = 𝐶 → (𝑧 = 𝑥𝐶 = 𝑥))
12 vex 3354 . . . . 5 𝑥 ∈ V
13 vex 3354 . . . . 5 𝑦 ∈ V
1412, 13br1steq 32008 . . . 4 (⟨𝑥, 𝑦⟩1st 𝑧𝑧 = 𝑥)
1510, 11, 14vtoclbg 3418 . . 3 (𝐶𝑋 → (⟨𝑥, 𝑦⟩1st 𝐶𝐶 = 𝑥))
165, 9, 15vtocl2g 3421 . 2 ((𝐴𝑉𝐵𝑊) → (𝐶𝑋 → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)))
17163impia 1109 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ⟨cop 4322   class class class wbr 4786  1st c1st 7313 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-1st 7315 This theorem is referenced by: (None)
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