Theorem 19.44v 2027

Description: Version of 19.44 2272 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 1911	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2		19.9v 2013	. . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓)
3	2	orbi2i 925	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))
4	1, 3	bitri 278	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 860  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996

This proof depends on definitions:  df-bi 210  df-or 861  df-ex 1809

This theorem is used by:  grothprim  10825