Theorem hbalw 1977

Description: Weak version of hbal 2036. Uses only Tarski's FOL axiom schemes. Unlike hbal 2036, this theorem requires that x and y be distinct, i.e. not be bundled. (Contributed by NM, 19-Apr-2017.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem hbalw

Step	Hyp	Ref	Expression
1		hbalw.2	. . 3 |- ( ph -> A. x ph )
2	1	alimi 1739	. 2 |- ( A. y ph -> A. y A. x ph )
3		hbalw.1	. . 3 |- ( x = z -> ( ph <-> ps ) )
4	3	alcomiw 1971	. 2 |- ( A. y A. x ph -> A. x A. y ph )
5	2, 4	syl 17	1 |- ( A. y ph -> A. x A. 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  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)