Theorem bj-eu3f 32017

Description: Version of eu3v 2486 where the dv condition is replaced with a non-freeness 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 2486. (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		eu5 2484	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2		bj-eu3f.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3	2	mo2 2467	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
4	3	anbi2i 726	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
5	1, 4	bitri 263	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699  ∃!weu 2458  ∃*wmo 2459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463

This theorem is referenced by: (None)