Theorem bj-ssblem2

Description: An instance of ax-11 proved without it. The converse may not be provable without ax-11 (since using alcomiw would require a DV on ph , x , which defeats the purpose). (Contributed by BJ, 22-Dec-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ssblem2	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑦 = 𝑡 → ( 𝑥 = 𝑦 → 𝜑 ) ) → ∀ 𝑦 ∀ 𝑥 ( 𝑦 = 𝑡 → ( 𝑥 = 𝑦 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		equequ1	⊢ ( 𝑦 = 𝑧 → ( 𝑦 = 𝑡 ↔ 𝑧 = 𝑡 ) )
2		equequ2	⊢ ( 𝑦 = 𝑧 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝑧 ) )
3	2	imbi1d	⊢ ( 𝑦 = 𝑧 → ( ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( 𝑥 = 𝑧 → 𝜑 ) ) )
4	1 3	imbi12d	⊢ ( 𝑦 = 𝑧 → ( ( 𝑦 = 𝑡 → ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ( 𝑧 = 𝑡 → ( 𝑥 = 𝑧 → 𝜑 ) ) ) )
5	4	alcomiw	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑦 = 𝑡 → ( 𝑥 = 𝑦 → 𝜑 ) ) → ∀ 𝑦 ∀ 𝑥 ( 𝑦 = 𝑡 → ( 𝑥 = 𝑦 → 𝜑 ) ) )