Theorem ax12

Description: Rederivation of Axiom ax-12 from ax12v (used only via sp ) , axc11r , and axc15 (on top of Tarski's FOL). Since this version depends on ax-13 , usage of the weaker ax12v , ax12w , ax12i are preferred. (Contributed by NM, 22-Jan-2007) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020) (Proof shortened by BJ, 4-Jul-2021) (New usage is discouraged.)

	Ref	Expression
Assertion	ax12	$$⊢x=y\to \left(\forall y\phi \to \forall x\left(x=y\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		axc11r	$$⊢\forall xx=y\to \left(\forall y\phi \to \forall x\phi \right)$$
2		ala1	$$⊢\forall x\phi \to \forall x\left(x=y\to \phi \right)$$
3	1 2	syl6	$$⊢\forall xx=y\to \left(\forall y\phi \to \forall x\left(x=y\to \phi \right)\right)$$
4	3	a1d	$$⊢\forall xx=y\to \left(x=y\to \left(\forall y\phi \to \forall x\left(x=y\to \phi \right)\right)\right)$$
5		sp	$$⊢\forall y\phi \to \phi$$
6		axc15	$$⊢\neg \forall xx=y\to \left(x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)\right)$$
7	5 6	syl7	$$⊢\neg \forall xx=y\to \left(x=y\to \left(\forall y\phi \to \forall x\left(x=y\to \phi \right)\right)\right)$$
8	4 7	pm2.61i	$$⊢x=y\to \left(\forall y\phi \to \forall x\left(x=y\to \phi \right)\right)$$