Theorem eu3 1024

Description: An alternate way of expressing existential uniqueness.

Hypothesis

Ref	Expression
eu3.1	⊢ (φ → ∀yφ)

Assertion

Ref	Expression
eu3	⊢ (∃!xφ ↔ (∃xφ ∧ ∃y∀x(φ → x = y)))

Distinct variable group(s):   x,y

Proof of Theorem eu3

Step	Hyp	Ref	Expression
1		eu3.1	. . 3 ⊢ (φ → ∀yφ)
2	1	eu2 1023	. 2 ⊢ (∃!xφ ↔ (∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)))
3	1	mo 1020	. . 3 ⊢ (∃y∀x(φ → x = y) ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))
4	3	anbi2i 367	. 2 ⊢ ((∃xφ ∧ ∃y∀x(φ → x = y)) ↔ (∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)))
5	2, 4	bitr4 154	1 ⊢ (∃!xφ ↔ (∃xφ ∧ ∃y∀x(φ → x = y)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  [wsb 852  ∃!weu 1007

This theorem is referenced by:  mo2 1026  eu5 1035  2eu4 1070  funeu 2685

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

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