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Theorem ax7w 1733
Description: Weak version of ax-7 1749 from which we can prove any ax-7 1749 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-7 1749, this theorem requires that  x and  y be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.)
Hypothesis
Ref Expression
ax7w.1  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ax7w  |-  ( 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 ax7w
StepHypRef Expression
1 ax7w.1 . 2  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
21alcomiw 1718 1  |-  ( A. x A. y ph  ->  A. y A. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551
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