Theorem moanim 2093

Description: Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 3-Dec-2001.)

Hypothesis

Ref	Expression
moanim.1	⊢ Ⅎ𝑥𝜑

Assertion

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

Proof of Theorem moanim

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anandi 585	. . . . 5 ⊢ ((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)))
2	1	imbi1i 237	. . . 4 ⊢ (((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦))
3		impexp 261	. . . 4 ⊢ (((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
4		sban 1948	. . . . . . 7 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
5		moanim.1	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜑
6	5	sbf 1770	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
7	6	anbi1i 455	. . . . . . 7 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ (𝜑 ∧ [𝑦 / 𝑥]𝜓))
8	4, 7	bitr2i 184	. . . . . 6 ⊢ ((𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ [𝑦 / 𝑥](𝜑 ∧ 𝜓))
9	8	anbi2i 454	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) ↔ ((𝜑 ∧ 𝜓) ∧ [𝑦 / 𝑥](𝜑 ∧ 𝜓)))
10	9	imbi1i 237	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (((𝜑 ∧ 𝜓) ∧ [𝑦 / 𝑥](𝜑 ∧ 𝜓)) → 𝑥 = 𝑦))
11	2, 3, 10	3bitr3i 209	. . 3 ⊢ ((𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (((𝜑 ∧ 𝜓) ∧ [𝑦 / 𝑥](𝜑 ∧ 𝜓)) → 𝑥 = 𝑦))
12	11	2albii 1464	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑥∀𝑦(((𝜑 ∧ 𝜓) ∧ [𝑦 / 𝑥](𝜑 ∧ 𝜓)) → 𝑥 = 𝑦))
13	5	19.21 1576	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
14		19.21v 1866	. . . 4 ⊢ (∀𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
15	14	albii 1463	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑥(𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
16		ax-17 1519	. . . . 5 ⊢ (𝜓 → ∀𝑦𝜓)
17	16	mo3h 2072	. . . 4 ⊢ (∃*𝑥𝜓 ↔ ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
18	17	imbi2i 225	. . 3 ⊢ ((𝜑 → ∃*𝑥𝜓) ↔ (𝜑 → ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
19	13, 15, 18	3bitr4ri 212	. 2 ⊢ ((𝜑 → ∃*𝑥𝜓) ↔ ∀𝑥∀𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
20		ax-17 1519	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑦(𝜑 ∧ 𝜓))
21	20	mo3h 2072	. 2 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥∀𝑦(((𝜑 ∧ 𝜓) ∧ [𝑦 / 𝑥](𝜑 ∧ 𝜓)) → 𝑥 = 𝑦))
22	12, 19, 21	3bitr4ri 212	1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346  Ⅎwnf 1453  [wsb 1755  ∃*wmo 2020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023

This theorem is referenced by:  moanimv  2094  moaneu  2095  moanmo  2096