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Theorem sbcaltop 34612
Description: Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
Assertion
Ref Expression
sbcaltop (𝐴 ∈ V → 𝐴 / 𝑥𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem sbcaltop
StepHypRef Expression
1 nfcsb1v 3881 . . . 4 𝑥𝐴 / 𝑥𝐶
2 nfcsb1v 3881 . . . 4 𝑥𝐴 / 𝑥𝐷
31, 2nfaltop 34611 . . 3 𝑥𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷
43a1i 11 . 2 (𝐴 ∈ V → 𝑥𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
5 csbeq1a 3870 . . . 4 (𝑥 = 𝐴𝐶 = 𝐴 / 𝑥𝐶)
6 altopeq1 34594 . . . 4 (𝐶 = 𝐴 / 𝑥𝐶 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐷⟫)
75, 6syl 17 . . 3 (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐷⟫)
8 csbeq1a 3870 . . . 4 (𝑥 = 𝐴𝐷 = 𝐴 / 𝑥𝐷)
9 altopeq2 34595 . . . 4 (𝐷 = 𝐴 / 𝑥𝐷 → ⟪𝐴 / 𝑥𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
108, 9syl 17 . . 3 (𝑥 = 𝐴 → ⟪𝐴 / 𝑥𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
117, 10eqtrd 2773 . 2 (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
124, 11csbiegf 3890 1 (𝐴 ∈ V → 𝐴 / 𝑥𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wnfc 2884  Vcvv 3444  csb 3856  caltop 34587
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 5257  ax-nul 5264  ax-pr 5385
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-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-sn 4588  df-pr 4590  df-altop 34589
This theorem is referenced by: (None)
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