Theorem 2reu5a 3685

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 3346	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑))
2		reu5 3346	. . . 4 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑))
3	2	rexbii 3086	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑))
4	2	rmobii 3352	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃*𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑))
5	3, 4	anbi12i 634	. 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 276	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 207   ∧ wa 396  ∃wrex 3063  ∃!wreu 3342  ∃*wrmo 3343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-mo 2543  df-eu 2573  df-rex 3064  df-rmo 3344  df-reu 3345

This theorem is referenced by:  2reu1  3829