Theorem sbcid 3067

Description: An identity theorem for substitution. See sbid 1827. (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 3055	. 2 |- ( [ x / x ] ph <-> [. x / x ]. ph )
2		sbid 1827	. 2 |- ( [ x / x ] ph <-> ph )
3	1, 2	bitr3i 186	1 |- ( [. x / x ]. ph <-> ph )

Syntax hints:   <-> wb 105   [wsb 1815   [.wsbc 3051

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052

This theorem is referenced by:  csbid  3155