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Theorem sbcopeq1a 8035
Description: Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a 3789 that avoids the existential quantifiers of copsexg 5492). (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 3479 . . . . 5 𝑥 ∈ V
2 vex 3479 . . . . 5 𝑦 ∈ V
31, 2op2ndd 7986 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = 𝑦)
43eqcomd 2739 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd𝐴))
5 sbceq1a 3789 . . 3 (𝑦 = (2nd𝐴) → (𝜑[(2nd𝐴) / 𝑦]𝜑))
64, 5syl 17 . 2 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑[(2nd𝐴) / 𝑦]𝜑))
71, 2op1std 7985 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝑥)
87eqcomd 2739 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1st𝐴))
9 sbceq1a 3789 . . 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 280 1 (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st𝐴) / 𝑥][(2nd𝐴) / 𝑦]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  [wsbc 3778  cop 4635  cfv 6544  1st c1st 7973  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975  df-2nd 7976
This theorem is referenced by:  dfopab2  8038  dfoprab3s  8039  ralxpes  8122  frpoins3xpg  8126
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