Theorem eu4 1036

Description: Uniqueness using implicit substitution.

Hypothesis

Ref	Expression
eu4.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
eu4	⊢ (∃!xφ ↔ (∃xφ ∧ ∀x∀y((φ ∧ ψ) → x = y)))

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

Proof of Theorem eu4

Step	Hyp	Ref	Expression
1		eu5 1035	. 2 ⊢ (∃!xφ ↔ (∃xφ ∧ ∃*xφ))
2		eu4.1	. . . 4 ⊢ (x = y → (φ ↔ ψ))
3	2	mo4 1029	. . 3 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ ψ) → x = y))
4	3	anbi2i 367	. 2 ⊢ ((∃xφ ∧ ∃*xφ) ↔ (∃xφ ∧ ∀x∀y((φ ∧ ψ) → x = y)))
5	1, 4	bitr 151	1 ⊢ (∃!xφ ↔ (∃xφ ∧ ∀x∀y((φ ∧ ψ) → x = y)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007  ∃*wmo 1008

This theorem is referenced by:  eueq 1427  hlimeu 5146

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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