Theorem moanimlem 2649

Description: Factor out the common proof skeleton of moanimv 2650 and moanim 2651. (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 2572	. . . 4 ⊢ (¬ ∃𝑥(𝜑 ∧ 𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))
5	3, 4	nsyl5 160	. . 3 ⊢ (¬ 𝜑 → ∃*𝑥(𝜑 ∧ 𝜓))
6		moan 2583	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∃wex 1812  ∃*wmo 2568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2570

This theorem is used by:  moanimv  2650  moanim  2651