Theorem moanim 2015

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 554	. . . . 5 ⊢ ((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)))
2	1	imbi1i 236	. . . 4 ⊢ (((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦))
3		impexp 259	. . . 4 ⊢ (((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
4		sban 1870	. . . . . . 7 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
5		moanim.1	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜑
6	5	sbf 1700	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
7	6	anbi1i 445	. . . . . . 7 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ (𝜑 ∧ [𝑦 / 𝑥]𝜓))
8	4, 7	bitr2i 183	. . . . . 6 ⊢ ((𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ [𝑦 / 𝑥](𝜑 ∧ 𝜓))
9	8	anbi2i 444	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) ↔ ((𝜑 ∧ 𝜓) ∧ [𝑦 / 𝑥](𝜑 ∧ 𝜓)))
10	9	imbi1i 236	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (((𝜑 ∧ 𝜓) ∧ [𝑦 / 𝑥](𝜑 ∧ 𝜓)) → 𝑥 = 𝑦))
11	2, 3, 10	3bitr3i 208	. . 3 ⊢ ((𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (((𝜑 ∧ 𝜓) ∧ [𝑦 / 𝑥](𝜑 ∧ 𝜓)) → 𝑥 = 𝑦))
12	11	2albii 1400	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑥∀𝑦(((𝜑 ∧ 𝜓) ∧ [𝑦 / 𝑥](𝜑 ∧ 𝜓)) → 𝑥 = 𝑦))
13	5	19.21 1515	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
14		19.21v 1794	. . . 4 ⊢ (∀𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
15	14	albii 1399	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑥(𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
16		ax-17 1459	. . . . 5 ⊢ (𝜓 → ∀𝑦𝜓)
17	16	mo3h 1994	. . . 4 ⊢ (∃*𝑥𝜓 ↔ ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
18	17	imbi2i 224	. . 3 ⊢ ((𝜑 → ∃*𝑥𝜓) ↔ (𝜑 → ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
19	13, 15, 18	3bitr4ri 211	. 2 ⊢ ((𝜑 → ∃*𝑥𝜓) ↔ ∀𝑥∀𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
20		ax-17 1459	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑦(𝜑 ∧ 𝜓))
21	20	mo3h 1994	. 2 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥∀𝑦(((𝜑 ∧ 𝜓) ∧ [𝑦 / 𝑥](𝜑 ∧ 𝜓)) → 𝑥 = 𝑦))
22	12, 19, 21	3bitr4ri 211	1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282  Ⅎwnf 1389  [wsb 1685  ∃*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  moanimv  2016  moaneu  2017  moanmo  2018