Theorem 19.44 2237

Description: Theorem 19.44 of [Margaris] p. 90. See 19.44v 1999 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)

Hypothesis

Ref	Expression
19.44.1	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
19.44	⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))

Proof of Theorem 19.44

Step	Hyp	Ref	Expression
1		19.43 1883	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2		19.44.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	19.9 2203	. . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓)
4	3	orbi2i 910	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))
5	1, 4	bitri 278	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))

Syntax hints:   ↔ wb 209   ∨ wo 844  ∃wex 1781  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-or 845  df-ex 1782  df-nf 1786

This theorem is referenced by:  eeor  2343