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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	$⊢ x = y \to (φ \leftrightarrow ψ)$
	ax12w.2	$⊢ y = z \to (φ \leftrightarrow χ)$
Assertion	ax12w	$⊢ x = y \to (\forall y φ \to \forall x (x = y \to φ))$

Proof

Step	Hyp	Ref	Expression
1		ax12w.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
2		ax12w.2	$⊢ y = z \to (φ \leftrightarrow χ)$
3	2	spw	$⊢ \forall y φ \to φ$
4	1	ax12wlem	$⊢ x = y \to (φ \to \forall x (x = y \to φ))$
5	3 4	syl5	$⊢ x = y \to (\forall y φ \to \forall x (x = y \to φ))$