Theorem moimv 1966

Description: Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)

Assertion

Ref	Expression
moimv	⊢ (∃*𝑥(𝜑 → 𝜓) → (𝜑 → ∃*𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem moimv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1 5	. . . . . . 7 ⊢ (𝜓 → (𝜑 → 𝜓))
2	1	a1i 9	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜑 → 𝜓)))
3	2	sbimi 1647	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → (𝜑 → 𝜓)))
4		nfv 1421	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
5	4	sbf 1660	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
6		sbim 1827	. . . . . . 7 ⊢ ([𝑦 / 𝑥](𝜓 → (𝜑 → 𝜓)) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑 → 𝜓)))
7	3, 5, 6	3imtr3i 189	. . . . . 6 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑 → 𝜓)))
8	2, 7	anim12d 318	. . . . 5 ⊢ (𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → ((𝜑 → 𝜓) ∧ [𝑦 / 𝑥](𝜑 → 𝜓))))
9	8	imim1d 69	. . . 4 ⊢ (𝜑 → ((((𝜑 → 𝜓) ∧ [𝑦 / 𝑥](𝜑 → 𝜓)) → 𝑥 = 𝑦) → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
10	9	2alimdv 1761	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(((𝜑 → 𝜓) ∧ [𝑦 / 𝑥](𝜑 → 𝜓)) → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
11		ax-17 1419	. . . 4 ⊢ ((𝜑 → 𝜓) → ∀𝑦(𝜑 → 𝜓))
12	11	mo3h 1953	. . 3 ⊢ (∃*𝑥(𝜑 → 𝜓) ↔ ∀𝑥∀𝑦(((𝜑 → 𝜓) ∧ [𝑦 / 𝑥](𝜑 → 𝜓)) → 𝑥 = 𝑦))
13		ax-17 1419	. . . 4 ⊢ (𝜓 → ∀𝑦𝜓)
14	13	mo3h 1953	. . 3 ⊢ (∃*𝑥𝜓 ↔ ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
15	10, 12, 14	3imtr4g 194	. 2 ⊢ (𝜑 → (∃*𝑥(𝜑 → 𝜓) → ∃*𝑥𝜓))
16	15	com12 27	1 ⊢ (∃*𝑥(𝜑 → 𝜓) → (𝜑 → ∃*𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  [wsb 1645  ∃*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by: (None)