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Theorem sbcaltop 32415
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 3690 . . . 4 𝑥𝐴 / 𝑥𝐶
2 nfcsb1v 3690 . . . 4 𝑥𝐴 / 𝑥𝐷
31, 2nfaltop 32414 . . 3 𝑥𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷
43a1i 11 . 2 (𝐴 ∈ V → 𝑥𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
5 csbeq1a 3683 . . . 4 (𝑥 = 𝐴𝐶 = 𝐴 / 𝑥𝐶)
6 altopeq1 32397 . . . 4 (𝐶 = 𝐴 / 𝑥𝐶 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐷⟫)
75, 6syl 17 . . 3 (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐷⟫)
8 csbeq1a 3683 . . . 4 (𝑥 = 𝐴𝐷 = 𝐴 / 𝑥𝐷)
9 altopeq2 32398 . . . 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 2794 . 2 (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
124, 11csbiegf 3698 1 (𝐴 ∈ V → 𝐴 / 𝑥𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wnfc 2889  Vcvv 3340  csb 3674  caltop 32390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-sn 4322  df-pr 4324  df-altop 32392
This theorem is referenced by: (None)
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