Theorem alcomiw 1971

Description: Weak version of alcom 2037. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)

Hypothesis

Ref	Expression
alcomiw.1	|- ( y = z -> ( ph <-> ps ) )

Assertion

Ref	Expression
alcomiw	|- ( A. x A. y ph -> A. y A. x ph )

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

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

Proof of Theorem alcomiw

Step	Hyp	Ref	Expression
1		alcomiw.1	. . . . 5 |- ( y = z -> ( ph <-> ps ) )
2	1	biimpd 219	. . . 4 |- ( y = z -> ( ph -> ps ) )
3	2	cbvalivw 1934	. . 3 |- ( A. y ph -> A. z ps )
4	3	alimi 1739	. 2 |- ( A. x A. y ph -> A. x A. z ps )
5		ax-5 1839	. 2 |- ( A. x A. z ps -> A. y A. x A. z ps )
6	1	biimprd 238	. . . . . 6 |- ( y = z -> ( ps -> ph ) )
7	6	equcoms 1947	. . . . 5 |- ( z = y -> ( ps -> ph ) )
8	7	spimvw 1927	. . . 4 |- ( A. z ps -> ph )
9	8	alimi 1739	. . 3 |- ( A. x A. z ps -> A. x ph )
10	9	alimi 1739	. 2 |- ( A. y A. x A. z ps -> A. y A. x ph )
11	4, 5, 10	3syl 18	1 |- ( A. x A. y ph -> A. y A. x ph )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481

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

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  hbalw  1977  ax11w  2007  bj-ssblem2  32631