Theorem 19.43 1884

Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 12-Mar-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 849	. . . 4 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
2	1	exbii 1850	. . 3 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ ∃𝑥(¬ 𝜑 → 𝜓))
3		19.35 1879	. . 3 ⊢ (∃𝑥(¬ 𝜑 → 𝜓) ↔ (∀𝑥 ¬ 𝜑 → ∃𝑥𝜓))
4		alnex 1783	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5	4	imbi1i 349	. . 3 ⊢ ((∀𝑥 ¬ 𝜑 → ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓))
6	2, 3, 5	3bitri 297	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓))
7		df-or 849	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓))
8	6, 7	bitr4i 278	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 848  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 207  df-or 849  df-ex 1782

This theorem is referenced by:  19.34  1994  19.44v  2000  19.45v  2001  19.44  2245  19.45  2246  eeor  2338  rexun  4136  uniprg  4866  uniun  4873  unopab  5165  zfpair  5363  dmun  5865  dmopab2rex  5872  coundi  6211  coundir  6212  kmlem16  10088  vdwapun  16945  satfdm  35551  satf0op  35559  dmopab3rexdif  35587  bj-nnfor  37053  bj-nnford  37054  bj-axseprep  37381  exor  43100  pm10.42  44791