Theorem ax11w 1721

Description: Weak version of ax-11 1746 from which we can prove any ax-11 1746 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that x and y be distinct (unless x does not occur in ph). (Contributed by NM, 10-Apr-2017.)

Hypotheses

Ref	Expression
ax11w.1	|- (x = y -> (ph <-> ps))
ax11w.2	|- (y = z -> (ph <-> ch))

Assertion

Ref	Expression
ax11w	|- (x = y -> (A.yph -> A.x(x = y -> ph)))

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

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

Proof of Theorem ax11w

Step	Hyp	Ref	Expression
1		ax11w.2	. . 3 |- (y = z -> (ph <-> ch))
2	1	spw 1694	. 2 |- (A.yph -> ph)
3		ax11w.1	. . 3 |- (x = y -> (ph <-> ps))
4	3	ax11wlem 1720	. 2 |- (x = y -> (ph -> A.x(x = y -> ph)))
5	2, 4	syl5 28	1 |- (x = y -> (A.yph -> A.x(x = y -> ph)))

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:  ax11wdemo  1723