Theorem 19.45 1731

Description: Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Hypothesis

Ref	Expression
19.45.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
19.45	⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Proof of Theorem 19.45

Step	Hyp	Ref	Expression
1		19.43 1677	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2		19.45.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	19.9 1693	. . 3 ⊢ (∃𝑥𝜑 ↔ 𝜑)
4	3	orbi1i 771	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
5	1, 4	bitri 184	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Syntax hints:   ↔ wb 105   ∨ wo 716  Ⅎwnf 1509  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-nf 1510

This theorem is referenced by:  eeor  1743