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Theorem csbeq2d 2953
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 2157 . . . 4 (𝜑 → (𝑦𝐵𝑦𝐶))
41, 3sbcbid 2894 . . 3 (𝜑 → ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2205 . 2 (𝜑 → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
6 df-csb 2932 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
7 df-csb 2932 . 2 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
85, 6, 73eqtr4g 2145 1 (𝜑𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wnf 1394  wcel 1438  {cab 2074  [wsbc 2838  csb 2931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-sbc 2839  df-csb 2932
This theorem is referenced by:  csbeq2dv  2954
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