Theorem sbcid 3047

Description: An identity theorem for substitution. See sbid 1822. (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 3035	. 2 |- ( [ x / x ] ph <-> [. x / x ]. ph )
2		sbid 1822	. 2 |- ( [ x / x ] ph <-> ph )
3	1, 2	bitr3i 186	1 |- ( [. x / x ]. ph <-> ph )

Syntax hints:   <-> wb 105   [wsb 1810   [.wsbc 3031

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-sbc 3032

This theorem is referenced by:  csbid  3135