Theorem ax11w 2007

Description: Weak version of ax-11 2034 from which we can prove any ax-11 2034 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2034, this theorem requires that x and y be distinct i.e. are not bundled. It is an alias of alcomiw 1971 introduced for labeling consistency. (Contributed by NM, 10-Apr-2017.) Use alcomiw 1971 instead. (New usage is discouraged.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem ax11w

Step	Hyp	Ref	Expression
1		ax11w.1	. 2 |- ( y = z -> ( ph <-> ps ) )
2	1	alcomiw 1971	1 |- ( A. x A. y ph -> A. y A. x 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  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)