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	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
	ax12w.2	$$⊢y=z\to \left(\phi \leftrightarrow \chi \right)$$
Assertion	ax12w	$$⊢x=y\to \left(\forall y\phi \to \forall x\left(x=y\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax12w.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		ax12w.2	$$⊢y=z\to \left(\phi \leftrightarrow \chi \right)$$
3	2	spw	$$⊢\forall y\phi \to \phi$$
4	1	ax12wlem	$$⊢x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$
5	3 4	syl5	$$⊢x=y\to \left(\forall y\phi \to \forall x\left(x=y\to \phi \right)\right)$$