Theorem bj-ax12ssb

Description: Axiom bj-ax12 expressed using substitution. (Contributed by BJ, 26-Dec-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ax12ssb	⊢ [ 𝑡 / 𝑥 ] ( 𝜑 → [ 𝑡 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		bj-ax12	⊢ ∀ 𝑥 ( 𝑥 = 𝑡 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
2		sb6	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) )
3	2	imbi2i	⊢ ( ( 𝜑 → [ 𝑡 / 𝑥 ] 𝜑 ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) )
4	3	imbi2i	⊢ ( ( 𝑥 = 𝑡 → ( 𝜑 → [ 𝑡 / 𝑥 ] 𝜑 ) ) ↔ ( 𝑥 = 𝑡 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) )
5	4	albii	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑡 → ( 𝜑 → [ 𝑡 / 𝑥 ] 𝜑 ) ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) )
6	1 5	mpbir	⊢ ∀ 𝑥 ( 𝑥 = 𝑡 → ( 𝜑 → [ 𝑡 / 𝑥 ] 𝜑 ) )
7		sb6	⊢ ( [ 𝑡 / 𝑥 ] ( 𝜑 → [ 𝑡 / 𝑥 ] 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → ( 𝜑 → [ 𝑡 / 𝑥 ] 𝜑 ) ) )
8	6 7	mpbir	⊢ [ 𝑡 / 𝑥 ] ( 𝜑 → [ 𝑡 / 𝑥 ] 𝜑 )