Theorem 19.45 2092

Description: Theorem 19.45 of [Margaris] p. 90. See 19.45v 1899 for a version requiring fewer axioms. (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 1798	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2		19.45.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	19.9 2058	. . 3 ⊢ (∃𝑥𝜑 ↔ 𝜑)
4	3	orbi1i 540	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
5	1, 4	bitri 262	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Syntax hints:   ↔ wb 194   ∨ wo 381  ∃wex 1694  Ⅎwnf 1698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032

This theorem depends on definitions:  df-bi 195  df-or 383  df-ex 1695  df-nf 1700

This theorem is referenced by:  eeor  2155