Theorem ax12i 1879

Description: Inference that has ax-12 2047 (without A. y) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 2047 in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)

Hypotheses

Ref	Expression
ax12i.1	|- ( x = y -> ( ph <-> ps ) )
ax12i.2	|- ( ps -> A. x ps )

Assertion

Ref	Expression
ax12i	|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Proof of Theorem ax12i

Step	Hyp	Ref	Expression
1		ax12i.1	. 2 |- ( x = y -> ( ph <-> ps ) )
2		ax12i.2	. . 3 |- ( ps -> A. x ps )
3	1	biimprcd 240	. . 3 |- ( ps -> ( x = y -> ph ) )
4	2, 3	alrimih 1751	. 2 |- ( ps -> A. x ( x = y -> ph ) )
5	1, 4	syl6bi 243	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

This theorem depends on definitions:  df-bi 197

This theorem is referenced by:  ax12wlem  2009