Theorem ax11-pm

Description: Proof of ax-11 similar to PM's proof of alcom (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 . Axiom ax-11 is used in the proof only through nfa2 . (Contributed by BJ, 15-Sep-2018) (Proof modification is discouraged.)

	Ref	Expression
Assertion	ax11-pm	$$⊢\forall x\forall y\phi \to \forall y\forall x\phi$$

Proof

Step	Hyp	Ref	Expression
1		2sp	$$⊢\forall x\forall y\phi \to \phi$$
2	1	gen2	$$⊢\forall y\forall x\left(\forall x\forall y\phi \to \phi \right)$$
3		nfa2	$$⊢Ⅎy\forall x\forall y\phi$$
4		nfa1	$$⊢Ⅎx\forall x\forall y\phi$$
5	3 4	2stdpc5	$$⊢\forall y\forall x\left(\forall x\forall y\phi \to \phi \right)\to \left(\forall x\forall y\phi \to \forall y\forall x\phi \right)$$
6	2 5	ax-mp	$$⊢\forall x\forall y\phi \to \forall y\forall x\phi$$