Theorem 2reu5a 41177

Description: Double restricted existential uniqueness in terms of restricted existence and restricted "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
2reu5a	⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑) ∧ ∃*𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑)))

Proof of Theorem 2reu5a

Step	Hyp	Ref	Expression
1		reu5 3159	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑))
2		reu5 3159	. . . 4 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑))
3	2	rexbii 3041	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑))
4	2	rmobii 3133	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃*𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑))
5	3, 4	anbi12i 733	. 2 ⊢ ((∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑) ∧ ∃*𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑)))
6	1, 5	bitri 264	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑) ∧ ∃*𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑)))

Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wrex 2913  ∃!wreu 2914  ∃*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-rex 2918  df-reu 2919  df-rmo 2920

This theorem is referenced by:  2reu1  41186