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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 ) ) )