Theorem 19.35 1907

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 43	. . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
2	1	aleximi 1862	. . 3 ⊢ (∀𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓))
3	2	com12 33	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
4		exnal 1857	. . . 4 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
5		pm2.21 124	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
6	5	eximi 1865	. . . 4 ⊢ (∃𝑥 ¬ 𝜑 → ∃𝑥(𝜑 → 𝜓))
7	4, 6	sylbir 238	. . 3 ⊢ (¬ ∀𝑥𝜑 → ∃𝑥(𝜑 → 𝜓))
8		exa1 1868	. . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 → 𝜓))
9	7, 8	ja 188	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))
10	3, 9	impbii 212	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀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-ex 1810

This theorem is referenced by:  19.35i  1908  19.35ri  1909  19.25  1910  19.43  1912  nfimd  1924  19.37imv  1977  speimfwALT  1994  19.39  2020  19.24  2021  19.36v  2023  19.37v  2027  19.36  2266  19.37  2268  spimt  2418  grothprim  10814  bj-nnfim  37377  bj-spimt2  37420  bj-spimtv  37429