Theorem eumo0 1022

Description: Existential uniqueness implies "at most one".

Hypothesis

Ref	Expression
eumo0.1	⊢ (φ → ∀yφ)

Assertion

Ref	Expression
eumo0	⊢ (∃!xφ → ∃y∀x(φ → x = y))

Distinct variable group(s):   x,y

Proof of Theorem eumo0

Step	Hyp	Ref	Expression
1		eumo0.1	. . 3 ⊢ (φ → ∀yφ)
2	1	euf 1011	. 2 ⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))
3		bi1 130	. . . 4 ⊢ ((φ ↔ x = y) → (φ → x = y))
4	3	19.20i 691	. . 3 ⊢ (∀x(φ ↔ x = y) → ∀x(φ → x = y))
5	4	19.22i 723	. 2 ⊢ (∃y∀x(φ ↔ x = y) → ∃y∀x(φ → x = y))
6	2, 5	sylbi 174	1 ⊢ (∃!xφ → ∃y∀x(φ → x = y))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007

This theorem is referenced by:  eu2 1023  mo2 1026

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-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-eu 1009

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