Theorem moimv 1044

Description: Move antecedent outside of "at most one".

Assertion

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

Distinct variable group(s):   φ,x

Proof of Theorem moimv

Step	Hyp	Ref	Expression
1		ax-1 3	. . . . . . 7 ⊢ (ψ → (φ → ψ))
2	1	a1i 7	. . . . . 6 ⊢ (φ → (ψ → (φ → ψ)))
3	2	syl4d 28	. . . . 5 ⊢ (φ → (((φ → ψ) → x = y) → (ψ → x = y)))
4	3	19.20dv 946	. . . 4 ⊢ (φ → (∀x((φ → ψ) → x = y) → ∀x(ψ → x = y)))
5	4	19.22dv 947	. . 3 ⊢ (φ → (∃y∀x((φ → ψ) → x = y) → ∃y∀x(ψ → x = y)))
6		ax-17 925	. . . 4 ⊢ ((φ → ψ) → ∀y(φ → ψ))
7	6	mo2 1026	. . 3 ⊢ (∃*x(φ → ψ) ↔ ∃y∀x((φ → ψ) → x = y))
8		ax-17 925	. . . 4 ⊢ (ψ → ∀yψ)
9	8	mo2 1026	. . 3 ⊢ (∃*xψ ↔ ∃y∀x(ψ → x = y))
10	5, 7, 9	3imtr4g 426	. 2 ⊢ (φ → (∃*x(φ → ψ) → ∃*xψ))
11	10	com12 13	1 ⊢ (∃*x(φ → ψ) → (φ → ∃*xψ))

Syntax hints:   → wi 2  ∀wal 672  ∃wex 678   = weq 797  ∃*wmo 1008

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

metamath.org