Theorem sbid2 2133

Description: An identity law for substitution. (Contributed by NM, 5-Aug-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 2132	. 2 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )
2		sbid2.1	. . 3 |- F/ x ph
3	2	sbf 2075	. 2 |- ( [ y / x ] ph <-> ph )
4	1, 3	bitri 241	1 |- ( [ y / x ] [ x / y ] ph <-> ph )

Syntax hints:   <-> wb 177   F/wnf 1550   [wsb 1655

This theorem is referenced by:  sbco2  2135  sb5rf  2139  sb6rf  2140  sbid2v  2173

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656