Theorem moanimlem 2706

Description: Factor out the common proof skeleton of moanimv 2707 and moanim 2708. (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 242	. 2 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) → (𝜑 → ∃*𝑥𝜓))
3		moanimlem.2	. . . . 5 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)
4		nexmo 2606	. . . . 5 ⊢ (¬ ∃𝑥(𝜑 ∧ 𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))
5	3, 4	nsyl4 158	. . . 4 ⊢ (¬ ∃*𝑥(𝜑 ∧ 𝜓) → 𝜑)
6	5	con1i 147	. . 3 ⊢ (¬ 𝜑 → ∃*𝑥(𝜑 ∧ 𝜓))
7		moan 2621	. . 3 ⊢ (∃*𝑥𝜓 → ∃*𝑥(𝜑 ∧ 𝜓))
8	6, 7	ja 175	. 2 ⊢ ((𝜑 → ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))
9	2, 8	impbii 201	1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∃wex 1878  ∃*wmo 2603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-mo 2605

This theorem is referenced by:  moanimv  2707  moanim  2708