Theorem eu4 2649

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 2603	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2		eu4.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	mo4 2600	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
4	3	anbi2i 634	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
5	1, 4	bitri 278	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565  ∃wex 1806  ∃*wmo 2571  ∃!weu 2602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-mo 2573  df-eu 2603

This theorem is referenced by:  euind  3696  eqeuel  4328  uniintsn  4954  eusv1  5363  omeu  8570  eroveu  8810  climeu  15606  pceu  16906  initoeu2lem2  18072  psgneu  19576  gsumval3eu  19974  frgr3vlem2  30566  3vfriswmgrlem  30569  onvfowev  35499  unirep  38253  rlimdmafv  47803  rlimdmafv2  47884