Theorem sbceq2a 3597

Description: Equality theorem for class substitution. Class version of sbequ12r 2267. (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 3596	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	eqcoms 2778	. 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 1630  [wsbc 3585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-12 2202  ax-ext 2750

This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-sbc 3586

This theorem is referenced by:  tfindes  7208  rabssnn0fi  12992  indexa  33853  fdc  33866  fdc1  33867  alrimii  34249  tratrbVD  39613