Theorem bj-eu3f 36784

Description: Version of eu3v 2566 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 2566. (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 2565	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2		bj-eu3f.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3	2	mof 2559	. . 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 206   ∧ wa 395  ∀wal 1533  ∃wex 1774  Ⅎwnf 1778  ∃*wmo 2534  ∃!weu 2564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-10 2137  ax-11 2153  ax-12 2173

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1775  df-nf 1779  df-mo 2536  df-eu 2565

This theorem is referenced by: (None)