Theorem 19.43 1605

Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)

Assertion

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

Proof of Theorem 19.43

Step	Hyp	Ref	Expression
1		df-or 359	. . . 4 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
2	1	exbii 1582	. . 3 ⊢ (∃x(φ ∨ ψ) ↔ ∃x(¬ φ → ψ))
3		19.35 1600	. . 3 ⊢ (∃x(¬ φ → ψ) ↔ (∀x ¬ φ → ∃xψ))
4		alnex 1543	. . . 4 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
5	4	imbi1i 315	. . 3 ⊢ ((∀x ¬ φ → ∃xψ) ↔ (¬ ∃xφ → ∃xψ))
6	2, 3, 5	3bitri 262	. 2 ⊢ (∃x(φ ∨ ψ) ↔ (¬ ∃xφ → ∃xψ))
7		df-or 359	. 2 ⊢ ((∃xφ ∨ ∃xψ) ↔ (¬ ∃xφ → ∃xψ))
8	6, 7	bitr4i 243	1 ⊢ (∃x(φ ∨ ψ) ↔ (∃xφ ∨ ∃xψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

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

This theorem is referenced by:  19.34  1663  19.44  1877  19.45  1878  rexun  3444  unipr  3906  uniun  3911  opeq  4620  unopab  4639  dmun  4913  coundi  5083  coundir  5084