Theorem dfeu 2595

Description: Rederive df-eu 2569 from the old definition eu6 2574. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 25-May-2019.) (Proof shortened by BJ, 7-Oct-2022.) (Proof modification is discouraged.) Use df-eu 2569 instead. (New usage is discouraged.)

Assertion

Ref	Expression
dfeu	⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))

Proof of Theorem dfeu

Step	Hyp	Ref	Expression
1		abai 823	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃!𝑥𝜑) ↔ (∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃!𝑥𝜑)))
2		euex 2577	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
3	2	pm4.71ri 560	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃!𝑥𝜑))
4		moeu 2583	. . 3 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
5	4	anbi2i 622	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃!𝑥𝜑)))
6	1, 3, 5	3bitr4i 302	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∃wex 1783  ∃*wmo 2538  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-mo 2540  df-eu 2569

This theorem is referenced by: (None)