Theorem sbcom4 2162

Description: Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2163 but may still be useful. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.)

Assertion

Ref	Expression
sbcom4	|- ( [ w / x ] [ y / z ] ph <-> [ y / x ] [ w / z ] ph )

Distinct variable groups:   ph, x, y, z   x, w, z

Allowed substitution hint:   ph( w)

Proof of Theorem sbcom4

Step	Hyp	Ref	Expression
1		nfv 1674	. . . . 5 |- F/ z ph
2	1	sbf 2081	. . . 4 |- ( [ y / z ] ph <-> ph )
3	2	sbbii 1709	. . 3 |- ( [ w / x ] [ y / z ] ph <-> [ w / x ] ph )
4		nfv 1674	. . . 4 |- F/ x ph
5	4	sbf 2081	. . 3 |- ( [ w / x ] ph <-> ph )
6	3, 5	bitri 249	. 2 |- ( [ w / x ] [ y / z ] ph <-> ph )
7	1	sbf 2081	. . . 4 |- ( [ w / z ] ph <-> ph )
8	7	sbbii 1709	. . 3 |- ( [ y / x ] [ w / z ] ph <-> [ y / x ] ph )
9	4	sbf 2081	. . 3 |- ( [ y / x ] ph <-> ph )
10	8, 9	bitri 249	. 2 |- ( [ y / x ] [ w / z ] ph <-> ph )
11	6, 10	bitr4i 252	1 |- ( [ w / x ] [ y / z ] ph <-> [ y / x ] [ w / z ] ph )

Syntax hints:   <-> wb 184   [wsb 1702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1955

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703

This theorem is referenced by: (None)