Theorem bj-ax12

Description: Remove a DV condition from bj-ax12v (using core axioms only). (Contributed by BJ, 26-Dec-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ax12	⊢ ∀ 𝑥 ( 𝑥 = 𝑡 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-ax12v	⊢ ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
2		equequ2	⊢ ( 𝑦 = 𝑡 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝑡 ) )
3	2	imbi1d	⊢ ( 𝑦 = 𝑡 → ( ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( 𝑥 = 𝑡 → 𝜑 ) ) )
4	3	albidv	⊢ ( 𝑦 = 𝑡 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
5	4	imbi2d	⊢ ( 𝑦 = 𝑡 → ( ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) )
6	2 5	imbi12d	⊢ ( 𝑦 = 𝑡 → ( ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) ↔ ( 𝑥 = 𝑡 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) ) )
7	6	albidv	⊢ ( 𝑦 = 𝑡 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) ) )
8	1 7	mpbii	⊢ ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑡 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) )
9		ax6ev	⊢ ∃ 𝑦 𝑦 = 𝑡
10	8 9	exlimiiv	⊢ ∀ 𝑥 ( 𝑥 = 𝑡 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )