Theorem sbcom2 2047

Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof modified to be intuitionistic by Jim Kingdon, 19-Feb-2018.)

Assertion

Ref	Expression
sbcom2	|- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph )

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

Allowed substitution hints:   ph( x, y, z, w)

Proof of Theorem sbcom2

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcom2v2 2046	. . . 4 |- ( [ v / z ] [ y / x ] ph <-> [ y / x ] [ v / z ] ph )
2	1	sbbii 1818	. . 3 |- ( [ w / v ] [ v / z ] [ y / x ] ph <-> [ w / v ] [ y / x ] [ v / z ] ph )
3		sbcom2v2 2046	. . 3 |- ( [ w / v ] [ y / x ] [ v / z ] ph <-> [ y / x ] [ w / v ] [ v / z ] ph )
4	2, 3	bitri 184	. 2 |- ( [ w / v ] [ v / z ] [ y / x ] ph <-> [ y / x ] [ w / v ] [ v / z ] ph )
5		ax-17 1579	. . 3 |- ( [ y / x ] ph -> A. v [ y / x ] ph )
6	5	sbco2vh 2005	. 2 |- ( [ w / v ] [ v / z ] [ y / x ] ph <-> [ w / z ] [ y / x ] ph )
7		ax-17 1579	. . . 4 |- ( ph -> A. v ph )
8	7	sbco2vh 2005	. . 3 |- ( [ w / v ] [ v / z ] ph <-> [ w / z ] ph )
9	8	sbbii 1818	. 2 |- ( [ y / x ] [ w / v ] [ v / z ] ph <-> [ y / x ] [ w / z ] ph )
10	4, 6, 9	3bitr3i 210	1 |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph )

Syntax hints:   <-> wb 105   [wsb 1815

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  2sb5rf  2049  2sb6rf  2050  sbco4lem  2066  sbco4  2067  sbmo  2146  cnvopab  5184