Theorem sbid2 2244

Description: An identity law for substitution. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sbid2.1	|- F/ x ph

Assertion

Ref	Expression
sbid2	|- ( [ y / x ] [ x / y ] ph <-> ph )

Proof of Theorem sbid2

Step	Hyp	Ref	Expression
1		sbco 2243	. 2 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )
2		sbid2.1	. . 3 |- F/ x ph
3	2	sbf 2211	. 2 |- ( [ y / x ] ph <-> ph )
4	1, 3	bitri 253	1 |- ( [ y / x ] [ x / y ] ph <-> ph )

Syntax hints:   <-> wb 188   F/wnf 1669   [wsb 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-12 1935  ax-13 2093

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-ex 1666  df-nf 1670  df-sb 1800

This theorem is referenced by:  sbtrt  2251  sbid2v  2288