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Theorem sbceq1dd 3726
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 3725 . 2 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜓))
41, 3mpbid 235 1 (𝜑[𝐵 / 𝑥]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  [wsbc 3720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-sbc 3721
This theorem is referenced by:  prmind2  16019  sdclem2  35180  sbceq1ddi  35561
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