Theorem alcomiw 1704

Description: Weak version of alcom 1737. 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.xA.yph -> A.yA.xph)

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 198	. . . 4 |- (y = z -> (ph -> ps))
3	2	cbvalivw 1674	. . 3 |- (A.yph -> A.zps)
4	3	alimi 1559	. 2 |- (A.xA.yph -> A.xA.zps)
5		ax-17 1616	. 2 |- (A.xA.zps -> A.yA.xA.zps)
6	1	biimprd 214	. . . . . 6 |- (y = z -> (ps -> ph))
7	6	equcoms 1681	. . . . 5 |- (z = y -> (ps -> ph))
8	7	spimvw 1669	. . . 4 |- (A.zps -> ph)
9	8	alimi 1559	. . 3 |- (A.xA.zps -> A.xph)
10	9	alimi 1559	. 2 |- (A.yA.xA.zps -> A.yA.xph)
11	4, 5, 10	3syl 18	1 |- (A.xA.yph -> A.yA.xph)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  hbalw  1709  ax7w  1718