Theorem moimv 1963

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 1644	. . . . . . 7 ⊢ ([y / x]φ → [y / x](ψ → (φ → ψ)))
4		nfv 1418	. . . . . . . 8 ⊢ Ⅎxφ
5	4	sbf 1657	. . . . . . 7 ⊢ ([y / x]φ ↔ φ)
6		sbim 1824	. . . . . . 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 1758	. . 3 ⊢ (φ → (∀x∀y(((φ → ψ) ∧ [y / x](φ → ψ)) → x = y) → ∀x∀y((ψ ∧ [y / x]ψ) → x = y)))
11		ax-17 1416	. . . 4 ⊢ ((φ → ψ) → ∀y(φ → ψ))
12	11	mo3h 1950	. . 3 ⊢ (∃*x(φ → ψ) ↔ ∀x∀y(((φ → ψ) ∧ [y / x](φ → ψ)) → x = y))
13		ax-17 1416	. . . 4 ⊢ (ψ → ∀yψ)
14	13	mo3h 1950	. . 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 1240  [wsb 1642  ∃*wmo 1898

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901

This theorem is referenced by: (None)