Theorem mo4f 1028

Description: "At most one" expressed using implicit substitution.

Hypotheses

Ref	Expression
mo4f.1	⊢ (ψ → ∀xψ)
mo4f.2	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
mo4f	⊢ (∃*xφ ↔ ∀x∀y((φ ∧ ψ) → x = y))

Distinct variable group(s):   x,y   φ,y

Proof of Theorem mo4f

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 ⊢ (φ → ∀yφ)
2	1	mo3 1027	. 2 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))
3		mo4f.1	. . . . . 6 ⊢ (ψ → ∀xψ)
4		mo4f.2	. . . . . 6 ⊢ (x = y → (φ ↔ ψ))
5	3, 4	sbie 904	. . . . 5 ⊢ ([y / x]φ ↔ ψ)
6	5	anbi2i 367	. . . 4 ⊢ ((φ ∧ [y / x]φ) ↔ (φ ∧ ψ))
7	6	imbi1i 161	. . 3 ⊢ (((φ ∧ [y / x]φ) → x = y) ↔ ((φ ∧ ψ) → x = y))
8	7	bi2al 696	. 2 ⊢ (∀x∀y((φ ∧ [y / x]φ) → x = y) ↔ ∀x∀y((φ ∧ ψ) → x = y))
9	2, 8	bitr 151	1 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ ψ) → x = y))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797  [wsb 852  ∃*wmo 1008

This theorem is referenced by:  mo4 1029  moop2 1910

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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