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Metamath Proof Explorer

Theorem bj-ax12w

Description: The general statement that ax12w proves. (Contributed by BJ, 20-Mar-2020)

Ref Expression
Hypotheses bj-ax12w.1 ⊢ φ → ψ ↔ χ
bj-ax12w.2 ⊢ y = z → ψ ↔ θ
Assertion bj-ax12w ⊢ φ → ∀ y ψ → ∀ x φ → ψ

Proof

Step Hyp Ref Expression
1 bj-ax12w.1 ⊢ φ → ψ ↔ χ
2 bj-ax12w.2 ⊢ y = z → ψ ↔ θ
3 2 spw ⊢ ∀ y ψ → ψ
4 1 bj-ax12wlem ⊢ φ → ψ → ∀ x φ → ψ
5 3 4 syl5 ⊢ φ → ∀ y ψ → ∀ x φ → ψ