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Metamath Proof Explorer

Theorem 19.37

Description: Theorem 19.37 of Margaris p. 90. See 19.37v for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993)

Ref Expression
Hypothesis 19.37.1 ⊢ Ⅎ 𝑥 𝜑
Assertion 19.37 ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∃ 𝑥 𝜓 ) )

Proof

Step Hyp Ref Expression
1 19.37.1 ⊢ Ⅎ 𝑥 𝜑
2 19.35 ⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
3 1 19.3 ⊢ ( ∀ 𝑥 𝜑 ↔ 𝜑 )
4 3 imbi1i ⊢ ( ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) ↔ ( 𝜑 → ∃ 𝑥 𝜓 ) )
5 2 4 bitri ⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∃ 𝑥 𝜓 ) )