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Theorem sbcaltop 35937
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 3940 . . . 4 𝑥𝐴 / 𝑥𝐶
2 nfcsb1v 3940 . . . 4 𝑥𝐴 / 𝑥𝐷
31, 2nfaltop 35936 . . 3 𝑥𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷
43a1i 11 . 2 (𝐴 ∈ V → 𝑥𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
5 csbeq1a 3929 . . . 4 (𝑥 = 𝐴𝐶 = 𝐴 / 𝑥𝐶)
6 altopeq1 35919 . . . 4 (𝐶 = 𝐴 / 𝑥𝐶 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐷⟫)
75, 6syl 17 . . 3 (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐷⟫)
8 csbeq1a 3929 . . . 4 (𝑥 = 𝐴𝐷 = 𝐴 / 𝑥𝐷)
9 altopeq2 35920 . . . 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 2774 . 2 (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
124, 11csbiegf 3949 1 (𝐴 ∈ V → 𝐴 / 𝑥𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2103  wnfc 2888  Vcvv 3482  csb 3915  caltop 35912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-sn 4649  df-pr 4651  df-altop 35914
This theorem is referenced by: (None)
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