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Metamath Proof Explorer

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 )