Theorem sbcid 3044

Description: An identity theorem for substitution. See sbid 1820. (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 3032	. 2 |- ( [ x / x ] ph <-> [. x / x ]. ph )
2		sbid 1820	. 2 |- ( [ x / x ] ph <-> ph )
3	1, 2	bitr3i 186	1 |- ( [. x / x ]. ph <-> ph )

Syntax hints:   <-> wb 105   [wsb 1808   [.wsbc 3028

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-sbc 3029

This theorem is referenced by:  csbid  3132