Theorem sbceq1 1443

Description: Equality theorem for class substitution.

Assertion

Ref	Expression
sbceq1	⊢ (x = A → (φ ↔ [A / x]φ))

Proof of Theorem sbceq1

Step	Hyp	Ref	Expression
1		dfsbcq 1442	. 2 ⊢ (x = A → ([x / x]φ ↔ [A / x]φ))
2		sbid 868	. 2 ⊢ ([x / x]φ ↔ φ)
3	1, 2	syl5bbr 412	1 ⊢ (x = A → (φ ↔ [A / x]φ))

Syntax hints:   → wi 2   ↔ wb 127  [wsb 852   = wceq 1091  [wsbc 1440

This theorem is referenced by:  sbc5g 1450  sbc6g 1451  sbc6 1453  elrabsf 1456  sbcel1 1466  sbcel2 1467  reuuni4 1959  nn1suc 4435  uzind 4603  nn0ind 4612

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-cleq 1097  df-clel 1099  df-sbc 1441

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