Theorem sbcid 3063

Description: An identity theorem for substitution. See sbid 1922. (Contributed by Mario Carneiro, 18-Feb-2017.)

Assertion

Ref	Expression
sbcid	⊢ ([̣x / x]̣φ ↔ φ)

Proof of Theorem sbcid

Step	Hyp	Ref	Expression
1		sbsbc 3051	. 2 ⊢ ([x / x]φ ↔ [̣x / x]̣φ)
2		sbid 1922	. 2 ⊢ ([x / x]φ ↔ φ)
3	1, 2	bitr3i 242	1 ⊢ ([̣x / x]̣φ ↔ φ)

Syntax hints:   ↔ wb 176  [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: (None)