Theorem 19.35 1877

Description: Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)

Assertion

Ref	Expression
19.35	⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem 19.35

Step	Hyp	Ref	Expression
1		pm2.27 42	. . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
2	1	aleximi 1832	. . 3 ⊢ (∀𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓))
3	2	com12 32	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
4		exnal 1827	. . . 4 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
5		pm2.21 123	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
6	5	eximi 1835	. . . 4 ⊢ (∃𝑥 ¬ 𝜑 → ∃𝑥(𝜑 → 𝜓))
7	4, 6	sylbir 235	. . 3 ⊢ (¬ ∀𝑥𝜑 → ∃𝑥(𝜑 → 𝜓))
8		exa1 1838	. . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 → 𝜓))
9	7, 8	ja 186	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))
10	3, 9	impbii 209	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by:  19.35i  1878  19.35ri  1879  19.25  1880  19.43  1882  nfimd  1894  19.37imv  1947  speimfwALT  1964  19.39  1984  19.24  1985  19.36v  1987  19.37v  1991  19.36  2230  19.37  2232  spimt  2390  grothprim  10848  bj-nfimt  36656  bj-nnfim  36764  bj-19.36im  36789  bj-19.37im  36790  bj-spimt2  36803  bj-spimtv  36812