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Theorem csbeq2d 3898
Description: Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
csbeq2d.1 𝑥𝜑
csbeq2d.2 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
csbeq2d (𝜑𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)

Proof of Theorem csbeq2d
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq2d.1 . . . 4 𝑥𝜑
2 csbeq2d.2 . . . . 5 (𝜑𝐵 = 𝐶)
32eleq2d 2815 . . . 4 (𝜑 → (𝑦𝐵𝑦𝐶))
41, 3sbcbid 3835 . . 3 (𝜑 → ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2797 . 2 (𝜑 → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
6 df-csb 3893 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
7 df-csb 3893 . 2 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
85, 6, 73eqtr4g 2793 1 (𝜑𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wnf 1778  wcel 2099  {cab 2705  [wsbc 3776  csb 3892
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-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-sbc 3777  df-csb 3893
This theorem is referenced by:  csbeq2dv  3899  poimirlem26  37119
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