Theorem mo4 1029

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

Hypothesis

Ref	Expression
mo4.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
mo4	⊢ (∃*xφ ↔ ∀x∀y((φ ∧ ψ) → x = y))

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

Proof of Theorem mo4

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (ψ → ∀xψ)
2		mo4.1	. 2 ⊢ (x = y → (φ ↔ ψ))
3	1, 2	mo4f 1028	1 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ ψ) → x = y))

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

This theorem is referenced by:  eu4 1036  supmo 2156  dffun3 2675  f1fv 2916  caoprmo 3084  th3qlem1 3250

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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