Theorem 19.45v 2032

Description: Version of 19.45 2277 with a disjoint variable 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 1915	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2		19.9v 2017	. . 3 ⊢ (∃𝑥𝜑 ↔ 𝜑)
3	2	orbi1i 927	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
4	1, 3	bitri 278	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813

This theorem is used by:  satfvsucsuc  35870  mosssn2  49628