Theorem 19.44 1660

Description: Theorem 19.44 of [Margaris] p. 90. (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 1607	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2		19.44.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	19.9 1623	. . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓)
4	3	orbi2i 751	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))
5	1, 4	bitri 183	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))

Syntax hints:   ↔ wb 104   ∨ wo 697  Ⅎwnf 1436  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  eeor  1673