Theorem sbceq2a 3447

Description: Equality theorem for class substitution. Class version of sbequ12r 2112. (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 3446	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	eqcoms 2630	. 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 1483  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by:  tfindes  7062  rabssnn0fi  12785  indexa  33528  fdc  33541  fdc1  33542  alrimii  33924  tratrbVD  39097