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Theorem sbceq1dd 2918
Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
Hypotheses
Ref Expression
sbceq1d.1 (𝜑𝐴 = 𝐵)
sbceq1dd.2 (𝜑[𝐴 / 𝑥]𝜓)
Assertion
Ref Expression
sbceq1dd (𝜑[𝐵 / 𝑥]𝜓)

Proof of Theorem sbceq1dd
StepHypRef Expression
1 sbceq1dd.2 . 2 (𝜑[𝐴 / 𝑥]𝜓)
2 sbceq1d.1 . . 3 (𝜑𝐴 = 𝐵)
32sbceq1d 2917 . 2 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜓))
41, 3mpbid 146 1 (𝜑[𝐵 / 𝑥]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  [wsbc 2912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136  df-sbc 2913
This theorem is referenced by:  prmind2  11835
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