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Theorem csbeq2d 2939
Description: Formula-building deduction rule 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 2152 . . . 4 (𝜑 → (𝑦𝐵𝑦𝐶))
41, 3sbcbid 2880 . . 3 (𝜑 → ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2200 . 2 (𝜑 → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
6 df-csb 2918 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
7 df-csb 2918 . 2 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
85, 6, 73eqtr4g 2140 1 (𝜑𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wnf 1390  wcel 1434  {cab 2069  [wsbc 2824  csb 2917
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-sbc 2825  df-csb 2918
This theorem is referenced by:  csbeq2dv  2940
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