Theorem sbcid 2779

Description: An identity theorem for substitution. See sbid 1657. (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 2768	. 2 |- ( [ x / x ] ph <-> [. x / x ]. ph )
2		sbid 1657	. 2 |- ( [ x / x ] ph <-> ph )
3	1, 2	bitr3i 175	1 |- ( [. x / x ]. ph <-> ph )

Syntax hints:   <-> wb 98   [wsb 1645   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765

This theorem is referenced by:  csbid  2859