Theorem sbcid 3013

Description: An identity theorem for substitution. See sbid 1796. (Contributed by Mario Carneiro, 18-Feb-2017.)

Assertion

Ref	Expression
sbcid	|- ( [. x / x ]. ph <-> ph )

Proof of Theorem sbcid

Step	Hyp	Ref	Expression
1		sbsbc 3001	. 2 |- ( [ x / x ] ph <-> [. x / x ]. ph )
2		sbid 1796	. 2 |- ( [ x / x ] ph <-> ph )
3	1, 2	bitr3i 186	1 |- ( [. x / x ]. ph <-> ph )

Syntax hints:   <-> wb 105   [wsb 1784   [.wsbc 2997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-sbc 2998

This theorem is referenced by:  csbid  3100