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Theorem sbcopeq1a 7111
 Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3413 that avoids the existential quantifiers of copsexg 4882). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
sbcopeq1a (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st𝐴) / 𝑥][(2nd𝐴) / 𝑦]𝜑𝜑))

Proof of Theorem sbcopeq1a
StepHypRef Expression
1 vex 3176 . . . . 5 𝑥 ∈ V
2 vex 3176 . . . . 5 𝑦 ∈ V
31, 2op2ndd 7070 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = 𝑦)
43eqcomd 2616 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd𝐴))
5 sbceq1a 3413 . . 3 (𝑦 = (2nd𝐴) → (𝜑[(2nd𝐴) / 𝑦]𝜑))
64, 5syl 17 . 2 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑[(2nd𝐴) / 𝑦]𝜑))
71, 2op1std 7069 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝑥)
87eqcomd 2616 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1st𝐴))
9 sbceq1a 3413 . . 3 (𝑥 = (1st𝐴) → ([(2nd𝐴) / 𝑦]𝜑[(1st𝐴) / 𝑥][(2nd𝐴) / 𝑦]𝜑))
108, 9syl 17 . 2 (𝐴 = ⟨𝑥, 𝑦⟩ → ([(2nd𝐴) / 𝑦]𝜑[(1st𝐴) / 𝑥][(2nd𝐴) / 𝑦]𝜑))
116, 10bitr2d 268 1 (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st𝐴) / 𝑥][(2nd𝐴) / 𝑦]𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  [wsbc 3402  ⟨cop 4131  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060 This theorem is referenced by:  dfopab2  7113  dfoprab3s  7114
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