Theorem bj-eu3f 36226

Description: Version of eu3v 2558 where the disjoint variable condition is replaced with a nonfreeness hypothesis. This is a "backup" of a theorem that used to be in the main part with label "eu3" and was deprecated in favor of eu3v 2558. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.)

Hypothesis

Ref	Expression
bj-eu3f.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
bj-eu3f	⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-eu3f

Step	Hyp	Ref	Expression
1		df-eu 2557	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2		bj-eu3f.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3	2	mof 2551	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
4	3	anbi2i 622	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
5	1, 4	bitri 275	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531  ∃wex 1773  Ⅎwnf 1777  ∃*wmo 2526  ∃!weu 2556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-mo 2528  df-eu 2557

This theorem is referenced by: (None)