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Theorem sbcopeq1a 8073
Description: Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a 3802 that avoids the existential quantifiers of copsexg 5502). (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 3482 . . . . 5 𝑥 ∈ V
2 vex 3482 . . . . 5 𝑦 ∈ V
31, 2op2ndd 8024 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = 𝑦)
43eqcomd 2741 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd𝐴))
5 sbceq1a 3802 . . 3 (𝑦 = (2nd𝐴) → (𝜑[(2nd𝐴) / 𝑦]𝜑))
64, 5syl 17 . 2 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑[(2nd𝐴) / 𝑦]𝜑))
71, 2op1std 8023 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝑥)
87eqcomd 2741 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1st𝐴))
9 sbceq1a 3802 . . 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 206   = wceq 1537  [wsbc 3791  cop 4637  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  dfopab2  8076  dfoprab3s  8077  ralxpes  8160  frpoins3xpg  8164
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