Theorem r19.43 2767

Description: Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
r19.43	⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ))

Proof of Theorem r19.43

Step	Hyp	Ref	Expression
1		r19.35 2759	. 2 ⊢ (∃x ∈ A (¬ φ → ψ) ↔ (∀x ∈ A ¬ φ → ∃x ∈ A ψ))
2		df-or 359	. . 3 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
3	2	rexbii 2640	. 2 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ ∃x ∈ A (¬ φ → ψ))
4		df-or 359	. . 3 ⊢ ((∃x ∈ A φ ∨ ∃x ∈ A ψ) ↔ (¬ ∃x ∈ A φ → ∃x ∈ A ψ))
5		ralnex 2625	. . . 4 ⊢ (∀x ∈ A ¬ φ ↔ ¬ ∃x ∈ A φ)
6	5	imbi1i 315	. . 3 ⊢ ((∀x ∈ A ¬ φ → ∃x ∈ A ψ) ↔ (¬ ∃x ∈ A φ → ∃x ∈ A ψ))
7	4, 6	bitr4i 243	. 2 ⊢ ((∃x ∈ A φ ∨ ∃x ∈ A ψ) ↔ (∀x ∈ A ¬ φ → ∃x ∈ A ψ))
8	1, 3, 7	3bitr4i 268	1 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357  ∀wral 2615  ∃wrex 2616

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-11 1746

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by:  r19.44av  2768  r19.45av  2769  r19.45zv  3648  iunun  4047  nncdiv3  6278