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	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		2sp	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → 𝜑 )
2	1	gen2	⊢ ∀ 𝑦 ∀ 𝑥 ( ∀ 𝑥 ∀ 𝑦 𝜑 → 𝜑 )
3		nfa2	⊢ Ⅎ 𝑦 ∀ 𝑥 ∀ 𝑦 𝜑
4		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 ∀ 𝑦 𝜑
5	3 4	2stdpc5	⊢ ( ∀ 𝑦 ∀ 𝑥 ( ∀ 𝑥 ∀ 𝑦 𝜑 → 𝜑 ) → ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) )
6	2 5	ax-mp	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )