Theorem eu4 2614

Description: Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Hypothesis

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

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem eu4

Step	Hyp	Ref	Expression
1		df-eu 2568	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2		eu4.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	mo4 2565	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
4	3	anbi2i 623	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
5	1, 4	bitri 275	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779  ∃*wmo 2537  ∃!weu 2567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2539  df-eu 2568

This theorem is referenced by:  euind  3707  eqeuel  4340  uniintsn  4961  eusv1  5361  omeu  8595  eroveu  8824  climeu  15569  pceu  16864  initoeu2lem2  18026  psgneu  19485  gsumval3eu  19883  frgr3vlem2  30201  3vfriswmgrlem  30204  unirep  37684  rlimdmafv  47154  rlimdmafv2  47235