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Metamath Proof Explorer

Theorem bj-ax12wlem

Description: A lemma used to prove a weak version of the axiom of substitution ax-12 . (Temporary comment: The general statement that ax12wlem proves.) (Contributed by BJ, 20-Mar-2020)

		Ref	Expression
	Hypothesis	bj-ax12wlem.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	Assertion	bj-ax12wlem	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-ax12wlem.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		ax-5	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
3	1 2	bj-ax12i	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )