Theorem 19.44v 1999

Description: Version of 19.44 2233 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)

Assertion

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

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.44v

Step	Hyp	Ref	Expression
1		19.43 1888	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2		19.9v 1990	. . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓)
3	2	orbi2i 909	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))
4	1, 3	bitri 274	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))

Syntax hints:   ↔ wb 205   ∨ wo 843  ∃wex 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974

This theorem depends on definitions:  df-bi 206  df-or 844  df-ex 1786

This theorem is referenced by:  grothprim  10574