Theorem sbceq2a 3816

Description: Equality theorem for class substitution. Class version of sbequ12r 2253. (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 3815	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	eqcoms 2748	. 2 ⊢ (𝐴 = 𝑥 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	bicomd 223	1 ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  [wsbc 3804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-sbc 3805

This theorem is referenced by:  ralrnmptw  7128  tfindes  7900  rabssnn0fi  14037  indexa  37693  fdc  37705  fdc1  37706  alrimii  38079  tratrbVD  44832