Theorem moanimlem 2648

Description: Factor out the common proof skeleton of moanimv 2649 and moanim 2650. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.) Factor out common proof lines. (Revised by Wolf Lammen, 8-Feb-2023.)

Hypotheses

Ref	Expression
moanimlem.1	⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓)))
moanimlem.2	⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)

Assertion

Ref	Expression
moanimlem	⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Proof of Theorem moanimlem

Step	Hyp	Ref	Expression
1		moanimlem.1	. . 3 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓)))
2	1	biimprcd 253	. 2 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) → (𝜑 → ∃*𝑥𝜓))
3		moanimlem.2	. . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)
4		nexmo 2571	. . . 4 ⊢ (¬ ∃𝑥(𝜑 ∧ 𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))
5	3, 4	nsyl5 160	. . 3 ⊢ (¬ 𝜑 → ∃*𝑥(𝜑 ∧ 𝜓))
6		moan 2582	. . 3 ⊢ (∃*𝑥𝜓 → ∃*𝑥(𝜑 ∧ 𝜓))
7	5, 6	ja 188	. 2 ⊢ ((𝜑 → ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))
8	2, 7	impbii 212	1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∃wex 1802  ∃*wmo 2567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-mo 2569

This theorem is referenced by:  moanimv  2649  moanim  2650