Theorem eu4 2676

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 2629	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2		eu4.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	mo4 2625	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
4	3	anbi2i 625	. 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 399  ∀wal 1536  ∃wex 1781  ∃*wmo 2596  ∃!weu 2628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2598  df-eu 2629

This theorem is referenced by:  euind  3663  eqeuel  4276  uniintsn  4875  eusv1  5257  omeu  8194  eroveu  8375  climeu  14904  pceu  16173  initoeu2lem2  17267  psgneu  18626  gsumval3eu  19017  frgr3vlem2  28059  3vfriswmgrlem  28062  unirep  35151  rlimdmafv  43733  rlimdmafv2  43814