Theorem sbceq2a 3441

Description: Equality theorem for class substitution. Class version of sbequ12r 2110. (Contributed by NM, 4-Jan-2017.)

Assertion

Ref	Expression
sbceq2a	⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Proof of Theorem sbceq2a

Step	Hyp	Ref	Expression
1		sbceq1a 3440	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	eqcoms 2628	. 2 ⊢ (𝐴 = 𝑥 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	bicomd 213	1 ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1481  [wsbc 3429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-12 2045  ax-ext 2600

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-sbc 3430

This theorem is referenced by:  tfindes  7047  rabssnn0fi  12768  indexa  33499  fdc  33512  fdc1  33513  alrimii  33895  tratrbVD  38917