Theorem moanim 1051

Description: Introduction of a conjunct into "at most one" quantifier.

Hypothesis

Ref	Expression
moanim.1	⊢ (φ → ∀xφ)

Assertion

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

Proof of Theorem moanim

Step	Hyp	Ref	Expression
1		impexp 276	. . . . 5 ⊢ (((φ ∧ ψ) → x = y) ↔ (φ → (ψ → x = y)))
2	1	bial 695	. . . 4 ⊢ (∀x((φ ∧ ψ) → x = y) ↔ ∀x(φ → (ψ → x = y)))
3		moanim.1	. . . . 5 ⊢ (φ → ∀xφ)
4	3	19.21 738	. . . 4 ⊢ (∀x(φ → (ψ → x = y)) ↔ (φ → ∀x(ψ → x = y)))
5	2, 4	bitr 151	. . 3 ⊢ (∀x((φ ∧ ψ) → x = y) ↔ (φ → ∀x(ψ → x = y)))
6	5	biex 733	. 2 ⊢ (∃y∀x((φ ∧ ψ) → x = y) ↔ ∃y(φ → ∀x(ψ → x = y)))
7		ax-17 925	. . 3 ⊢ ((φ ∧ ψ) → ∀y(φ ∧ ψ))
8	7	mo2 1026	. 2 ⊢ (∃*x(φ ∧ ψ) ↔ ∃y∀x((φ ∧ ψ) → x = y))
9		ax-17 925	. . . . 5 ⊢ (ψ → ∀yψ)
10	9	mo2 1026	. . . 4 ⊢ (∃*xψ ↔ ∃y∀x(ψ → x = y))
11	10	imbi2i 160	. . 3 ⊢ ((φ → ∃*xψ) ↔ (φ → ∃y∀x(ψ → x = y)))
12		19.37v 961	. . 3 ⊢ (∃y(φ → ∀x(ψ → x = y)) ↔ (φ → ∃y∀x(ψ → x = y)))
13	11, 12	bitr4 154	. 2 ⊢ ((φ → ∃*xψ) ↔ ∃y(φ → ∀x(ψ → x = y)))
14	6, 8, 13	3bitr4 158	1 ⊢ (∃*x(φ ∧ ψ) ↔ (φ → ∃*xψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  ∃*wmo 1008

This theorem is referenced by:  moanimv 1052  2eu1 1067

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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