Theorem mooran1 1049

Description: "At most one" imports disjunction to conjunction.

Assertion

Ref	Expression
mooran1	⊢ ((∃*xφ ∨ ∃*xψ) → ∃*x(φ ∧ ψ))

Proof of Theorem mooran1

Step	Hyp	Ref	Expression
1		moan 1046	. . 3 ⊢ (∃*xφ → ∃*x(ψ ∧ φ))
2		ancom 333	. . . 4 ⊢ ((ψ ∧ φ) ↔ (φ ∧ ψ))
3	2	bimo 1031	. . 3 ⊢ (∃*x(ψ ∧ φ) ↔ ∃*x(φ ∧ ψ))
4	1, 3	sylib 173	. 2 ⊢ (∃*xφ → ∃*x(φ ∧ ψ))
5		moan 1046	. 2 ⊢ (∃*xψ → ∃*x(φ ∧ ψ))
6	4, 5	jaoi 275	1 ⊢ ((∃*xφ ∨ ∃*xψ) → ∃*x(φ ∧ ψ))

Syntax hints:   → wi 2   ∨ wo 195   ∧ wa 196  ∃*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

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