Theorem 19.43 1912

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 861	. . . 4 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
2	1	exbii 1878	. . 3 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ ∃𝑥(¬ 𝜑 → 𝜓))
3		19.35 1907	. . 3 ⊢ (∃𝑥(¬ 𝜑 → 𝜓) ↔ (∀𝑥 ¬ 𝜑 → ∃𝑥𝜓))
4		alnex 1811	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5	4	imbi1i 352	. . 3 ⊢ ((∀𝑥 ¬ 𝜑 → ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓))
6	2, 3, 5	3bitri 300	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓))
7		df-or 861	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓))
8	6, 7	bitr4i 281	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 860  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1810

This theorem is referenced by:  19.34  2022  19.44v  2028  19.45v  2029  19.44  2273  19.45  2274  eeor  2366  rexun  4149  uniprg  4888  uniun  4895  unopab  5191  zfpair  5392  dmun  5900  dmopab2rex  5907  coundi  6248  coundir  6249  kmlem16  10145  vdwapun  17029  satfdm  35861  satf0op  35869  dmopab3rexdif  35897  bj-nnfor  37381  bj-nnford  37382  bj-axseprep  37711  exor  43399  pm10.42  45074