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Metamath Proof Explorer

Theorem bj-ax12i

Description: A weakening of bj-ax12ig that is sufficient to prove a weak form of the axiom of substitution ax-12 . The general statement of which ax12i is an instance. (Contributed by BJ, 29-Sep-2019)

	Ref	Expression
Hypotheses	bj-ax12i.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	bj-ax12i.2	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
Assertion	bj-ax12i	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-ax12i.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		bj-ax12i.2	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
3	2	a1i	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
4	1 3	bj-ax12ig	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )