Theorem 19.45 1878

Description: Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
19.45.1	⊢ Ⅎxφ

Assertion

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

Proof of Theorem 19.45

Step	Hyp	Ref	Expression
1		19.43 1605	. 2 ⊢ (∃x(φ ∨ ψ) ↔ (∃xφ ∨ ∃xψ))
2		19.45.1	. . . 4 ⊢ Ⅎxφ
3	2	19.9 1783	. . 3 ⊢ (∃xφ ↔ φ)
4	3	orbi1i 506	. 2 ⊢ ((∃xφ ∨ ∃xψ) ↔ (φ ∨ ∃xψ))
5	1, 4	bitri 240	1 ⊢ (∃x(φ ∨ ψ) ↔ (φ ∨ ∃xψ))

Syntax hints:   ↔ wb 176   ∨ wo 357  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  eeor  1885