Theorem ax12wlem 2009

Description: Lemma for weak version of ax-12 2047. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 2010. (Contributed by NM, 10-Apr-2017.)

Hypothesis

Ref	Expression
ax12wlemw.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
ax12wlem	|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Distinct variable group:   ps, x

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

Proof of Theorem ax12wlem

Step	Hyp	Ref	Expression
1		ax12wlemw.1	. 2 |- ( x = y -> ( ph <-> ps ) )
2		ax-5 1839	. 2 |- ( ps -> A. x ps )
3	1, 2	ax12i 1879	1 |- ( x = y -> ( ph -> A. x ( x = y -> 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

This theorem depends on definitions:  df-bi 197

This theorem is referenced by:  ax12w  2010