Theorem hbalw 1709

Description: Weak version of hbal 1736. Uses only Tarski's FOL axiom schemes. Unlike hbal 1736, this theorem requires that x and y be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.)

Hypotheses

Ref	Expression
hbalw.1	|- (x = z -> (ph <-> ps))
hbalw.2	|- (ph -> A.xph)

Assertion

Ref	Expression
hbalw	|- (A.yph -> A.xA.yph)

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

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

Proof of Theorem hbalw

Step	Hyp	Ref	Expression
1		hbalw.2	. . 3 |- (ph -> A.xph)
2	1	alimi 1559	. 2 |- (A.yph -> A.yA.xph)
3		hbalw.1	. . 3 |- (x = z -> (ph <-> ps))
4	3	alcomiw 1704	. 2 |- (A.yA.xph -> A.xA.yph)
5	2, 4	syl 15	1 |- (A.yph -> A.xA.yph)

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: (None)