Theorem alcomiw

Description: Weak version of ax-11 . See alcomw for the biconditional form. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017) (Proof shortened by Wolf Lammen, 28-Dec-2023)

	Ref	Expression
Hypothesis	alcomiw.1	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
Assertion	alcomiw	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		alcomiw.1	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
2	1	cbvalvw	⊢ ( ∀ 𝑦 𝜑 ↔ ∀ 𝑧 𝜓 )
3	2	biimpi	⊢ ( ∀ 𝑦 𝜑 → ∀ 𝑧 𝜓 )
4	3	alimi	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑧 𝜓 )
5		ax-5	⊢ ( ∀ 𝑥 ∀ 𝑧 𝜓 → ∀ 𝑦 ∀ 𝑥 ∀ 𝑧 𝜓 )
6	1	biimprd	⊢ ( 𝑦 = 𝑧 → ( 𝜓 → 𝜑 ) )
7	6	equcoms	⊢ ( 𝑧 = 𝑦 → ( 𝜓 → 𝜑 ) )
8	7	spimvw	⊢ ( ∀ 𝑧 𝜓 → 𝜑 )
9	8	2alimi	⊢ ( ∀ 𝑦 ∀ 𝑥 ∀ 𝑧 𝜓 → ∀ 𝑦 ∀ 𝑥 𝜑 )
10	4 5 9	3syl	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )