Theorem moimv 2111

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 6	. . . . . . 7 ⊢ (𝜓 → (𝜑 → 𝜓))
2	1	a1i 9	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜑 → 𝜓)))
3	2	sbimi 1778	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → (𝜑 → 𝜓)))
4		nfv 1542	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
5	4	sbf 1791	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
6		sbim 1972	. . . . . . 7 ⊢ ([𝑦 / 𝑥](𝜓 → (𝜑 → 𝜓)) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑 → 𝜓)))
7	3, 5, 6	3imtr3i 200	. . . . . 6 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑 → 𝜓)))
8	2, 7	anim12d 335	. . . . 5 ⊢ (𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → ((𝜑 → 𝜓) ∧ [𝑦 / 𝑥](𝜑 → 𝜓))))
9	8	imim1d 75	. . . 4 ⊢ (𝜑 → ((((𝜑 → 𝜓) ∧ [𝑦 / 𝑥](𝜑 → 𝜓)) → 𝑥 = 𝑦) → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
10	9	2alimdv 1895	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(((𝜑 → 𝜓) ∧ [𝑦 / 𝑥](𝜑 → 𝜓)) → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
11		ax-17 1540	. . . 4 ⊢ ((𝜑 → 𝜓) → ∀𝑦(𝜑 → 𝜓))
12	11	mo3h 2098	. . 3 ⊢ (∃*𝑥(𝜑 → 𝜓) ↔ ∀𝑥∀𝑦(((𝜑 → 𝜓) ∧ [𝑦 / 𝑥](𝜑 → 𝜓)) → 𝑥 = 𝑦))
13		ax-17 1540	. . . 4 ⊢ (𝜓 → ∀𝑦𝜓)
14	13	mo3h 2098	. . 3 ⊢ (∃*𝑥𝜓 ↔ ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
15	10, 12, 14	3imtr4g 205	. 2 ⊢ (𝜑 → (∃*𝑥(𝜑 → 𝜓) → ∃*𝑥𝜓))
16	15	com12 30	1 ⊢ (∃*𝑥(𝜑 → 𝜓) → (𝜑 → ∃*𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362  [wsb 1776  ∃*wmo 2046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049

This theorem is referenced by: (None)