Theorem ax7w 1718

Description: Weak version of ax-7 1734 from which we can prove any ax-7 1734 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-7 1734, this theorem requires that x and y be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.)

Hypothesis

Ref	Expression
ax7w.1	|- (y = z -> (ph <-> ps))

Assertion

Ref	Expression
ax7w	|- (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 ax7w

Step	Hyp	Ref	Expression
1		ax7w.1	. 2 |- (y = z -> (ph <-> ps))
2	1	alcomiw 1704	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: (None)