Theorem sbcid 2971

Description: An identity theorem for substitution. See sbid 1768. (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 2960	. 2 |- ( [ x / x ] ph <-> [. x / x ]. ph )
2		sbid 1768	. 2 |- ( [ x / x ] ph <-> ph )
3	1, 2	bitr3i 185	1 |- ( [. x / x ]. ph <-> ph )

Syntax hints:   <-> wb 104   [wsb 1756   [.wsbc 2956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1441  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-sbc 2957

This theorem is referenced by:  csbid  3058