Metamath Proof Explorer


Theorem ax12w

Description: Weak version of ax-12 from which we can prove any ax-12 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 ). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for ph , see ax12wdemo . (Contributed by NM, 10-Apr-2017)

Ref Expression
Hypotheses ax12w.1
|- ( x = y -> ( ph <-> ps ) )
ax12w.2
|- ( y = z -> ( ph <-> ch ) )
Assertion ax12w
|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )

Proof

Step Hyp Ref Expression
1 ax12w.1
 |-  ( x = y -> ( ph <-> ps ) )
2 ax12w.2
 |-  ( y = z -> ( ph <-> ch ) )
3 2 spw
 |-  ( A. y ph -> ph )
4 1 ax12wlem
 |-  ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
5 3 4 syl5
 |-  ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )