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Metamath Proof Explorer

Theorem 19.36v

Description: Version of 19.36 with a disjoint variable condition instead of a nonfreeness hypothesis. (Contributed by NM, 18-Aug-1993) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020)

		Ref	Expression
	Assertion	19.36v	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.35	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
2		19.9v	⊢ ( ∃ 𝑥 𝜓 ↔ 𝜓 )
3	2	imbi2i	⊢ ( ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → 𝜓 ) )
4	1 3	bitri	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → 𝜓 ) )