Theorem 19.45v 1913

Description: Version of 19.45 2107 with a dv condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)

Assertion

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

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.45v

Step	Hyp	Ref	Expression
1		19.43 1810	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2		19.9v 1896	. . 3 ⊢ (∃𝑥𝜑 ↔ 𝜑)
3	2	orbi1i 542	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
4	1, 3	bitri 264	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Syntax hints:   ↔ wb 196   ∨ wo 383  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705

This theorem is referenced by: (None)