Theorem sbcid 3452

Description: An identity theorem for substitution. See sbid 2114. (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 3439	. 2 |- ( [ x / x ] ph <-> [. x / x ]. ph )
2		sbid 2114	. 2 |- ( [ x / x ] ph <-> ph )
3	1, 2	bitr3i 266	1 |- ( [. x / x ]. ph <-> ph )

Syntax hints:   <-> wb 196   [wsb 1880   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by:  csbid  3541  snfil  21668  ex-natded9.26  27276  bnj605  30977  dedths  34248  frege93  38250