Theorem 19.43 1901

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 859	. . . 4 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
2	1	exbii 1867	. . 3 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ ∃𝑥(¬ 𝜑 → 𝜓))
3		19.35 1896	. . 3 ⊢ (∃𝑥(¬ 𝜑 → 𝜓) ↔ (∀𝑥 ¬ 𝜑 → ∃𝑥𝜓))
4		alnex 1800	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5	4	imbi1i 351	. . 3 ⊢ ((∀𝑥 ¬ 𝜑 → ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓))
6	2, 3, 5	3bitri 299	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓))
7		df-or 859	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓))
8	6, 7	bitr4i 280	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 858  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828

This theorem depends on definitions:  df-bi 209  df-or 859  df-ex 1799

This theorem is referenced by:  19.34  2011  19.44v  2017  19.45v  2018  19.44  2271  19.45  2272  eeor  2364  rexun  4146  uniprg  4878  uniun  4885  unopab  5177  zfpair  5375  dmun  5882  dmopab2rex  5889  coundi  6229  coundir  6230  kmlem16  10116  vdwapun  17001  satfdm  35680  satf0op  35688  dmopab3rexdif  35716  bj-nnfor  37192  bj-nnford  37193  bj-axseprep  37520  exor  43210  pm10.42  44901