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Metamath Proof Explorer

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 ) ) )