Theorem moanim 2260

Description: Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)

Hypothesis

Ref	Expression
moanim.1	⊢ Ⅎxφ

Assertion

Ref	Expression
moanim	⊢ (∃*x(φ ∧ ψ) ↔ (φ → ∃*xψ))

Proof of Theorem moanim

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		impexp 433	. . . . 5 ⊢ (((φ ∧ ψ) → x = y) ↔ (φ → (ψ → x = y)))
2	1	albii 1566	. . . 4 ⊢ (∀x((φ ∧ ψ) → x = y) ↔ ∀x(φ → (ψ → x = y)))
3		moanim.1	. . . . 5 ⊢ Ⅎxφ
4	3	19.21 1796	. . . 4 ⊢ (∀x(φ → (ψ → x = y)) ↔ (φ → ∀x(ψ → x = y)))
5	2, 4	bitri 240	. . 3 ⊢ (∀x((φ ∧ ψ) → x = y) ↔ (φ → ∀x(ψ → x = y)))
6	5	exbii 1582	. 2 ⊢ (∃y∀x((φ ∧ ψ) → x = y) ↔ ∃y(φ → ∀x(ψ → x = y)))
7		nfv 1619	. . 3 ⊢ Ⅎy(φ ∧ ψ)
8	7	mo2 2233	. 2 ⊢ (∃*x(φ ∧ ψ) ↔ ∃y∀x((φ ∧ ψ) → x = y))
9		nfv 1619	. . . . 5 ⊢ Ⅎyψ
10	9	mo2 2233	. . . 4 ⊢ (∃*xψ ↔ ∃y∀x(ψ → x = y))
11	10	imbi2i 303	. . 3 ⊢ ((φ → ∃*xψ) ↔ (φ → ∃y∀x(ψ → x = y)))
12		19.37v 1899	. . 3 ⊢ (∃y(φ → ∀x(ψ → x = y)) ↔ (φ → ∃y∀x(ψ → x = y)))
13	11, 12	bitr4i 243	. 2 ⊢ ((φ → ∃*xψ) ↔ ∃y(φ → ∀x(ψ → x = y)))
14	6, 8, 13	3bitr4i 268	1 ⊢ (∃*x(φ ∧ ψ) ↔ (φ → ∃*xψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  ∃*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  moanimv  2262  moaneu  2263  moanmo  2264  2eu1  2284