Theorem sbceq1a 3057

Description: Equality theorem for class substitution. Class version of sbequ12 1919. (Contributed by NM, 26-Sep-2003.)

Assertion

Ref	Expression
sbceq1a	⊢ (x = A → (φ ↔ [̣A / x]̣φ))

Proof of Theorem sbceq1a

Step	Hyp	Ref	Expression
1		sbid 1922	. 2 ⊢ ([x / x]φ ↔ φ)
2		dfsbcq2 3050	. 2 ⊢ (x = A → ([x / x]φ ↔ [̣A / x]̣φ))
3	1, 2	syl5bbr 250	1 ⊢ (x = A → (φ ↔ [̣A / x]̣φ))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648  [̣wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3048

This theorem is referenced by:  sbceq2a  3058  elrabsf  3085  cbvralcsf  3199