Theorem sbcid 2830

Description: An identity theorem for substitution. See sbid 1697. (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 2819	. 2 |- ( [ x / x ] ph <-> [. x / x ]. ph )
2		sbid 1697	. 2 |- ( [ x / x ] ph <-> ph )
3	1, 2	bitr3i 184	1 |- ( [. x / x ]. ph <-> ph )

Syntax hints:   <-> wb 103   [wsb 1685   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-sbc 2816

This theorem is referenced by:  csbid  2915