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Theorem sbcaltop 33858
 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 3831 . . . 4 𝑥𝐴 / 𝑥𝐶
2 nfcsb1v 3831 . . . 4 𝑥𝐴 / 𝑥𝐷
31, 2nfaltop 33857 . . 3 𝑥𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷
43a1i 11 . 2 (𝐴 ∈ V → 𝑥𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
5 csbeq1a 3821 . . . 4 (𝑥 = 𝐴𝐶 = 𝐴 / 𝑥𝐶)
6 altopeq1 33840 . . . 4 (𝐶 = 𝐴 / 𝑥𝐶 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐷⟫)
75, 6syl 17 . . 3 (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐷⟫)
8 csbeq1a 3821 . . . 4 (𝑥 = 𝐴𝐷 = 𝐴 / 𝑥𝐷)
9 altopeq2 33841 . . . 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 2793 . 2 (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
124, 11csbiegf 3840 1 (𝐴 ∈ V → 𝐴 / 𝑥𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2899  Vcvv 3409  ⦋csb 3807  ⟪caltop 33833 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-pr 4528  df-altop 33835 This theorem is referenced by: (None)
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