Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbcaltop Structured version   Visualization version   GIF version

Theorem sbcaltop 35567
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 3914 . . . 4 𝑥𝐴 / 𝑥𝐶
2 nfcsb1v 3914 . . . 4 𝑥𝐴 / 𝑥𝐷
31, 2nfaltop 35566 . . 3 𝑥𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷
43a1i 11 . 2 (𝐴 ∈ V → 𝑥𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
5 csbeq1a 3903 . . . 4 (𝑥 = 𝐴𝐶 = 𝐴 / 𝑥𝐶)
6 altopeq1 35549 . . . 4 (𝐶 = 𝐴 / 𝑥𝐶 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐷⟫)
75, 6syl 17 . . 3 (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐷⟫)
8 csbeq1a 3903 . . . 4 (𝑥 = 𝐴𝐷 = 𝐴 / 𝑥𝐷)
9 altopeq2 35550 . . . 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 2767 . 2 (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
124, 11csbiegf 3923 1 (𝐴 ∈ V → 𝐴 / 𝑥𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wnfc 2878  Vcvv 3469  csb 3889  caltop 35542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-sn 4625  df-pr 4627  df-altop 35544
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator