Theorem sbceq2a 3695

Description: Equality theorem for class substitution. Class version of sbequ12r 2252. (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 3694	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	eqcoms 2744	. 2 ⊢ (𝐴 = 𝑥 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	bicomd 226	1 ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1543  [wsbc 3683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-sbc 3684

This theorem is referenced by:  ralrnmptw  6891  tfindes  7619  rabssnn0fi  13524  indexa  35577  fdc  35589  fdc1  35590  alrimii  35963  tratrbVD  42095