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Metamath Proof Explorer

Theorem 19.45

Description: Theorem 19.45 of Margaris p. 90. See 19.45v for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993)

Ref Expression
Hypothesis 19.45.1 ⊢ Ⅎ 𝑥 𝜑
Assertion 19.45 ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∃ 𝑥 𝜓 ) )

Proof

Step Hyp Ref Expression
1 19.45.1 ⊢ Ⅎ 𝑥 𝜑
2 19.43 ⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) )
3 1 19.9 ⊢ ( ∃ 𝑥 𝜑 ↔ 𝜑 )
4 3 orbi1i ⊢ ( ( ∃ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) ↔ ( 𝜑 ∨ ∃ 𝑥 𝜓 ) )
5 2 4 bitri ⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∃ 𝑥 𝜓 ) )