Theorem 2reu5lem2 3744

Description: Lemma for 2reu5 3746. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
2reu5lem2	⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem 2reu5lem2

Step	Hyp	Ref	Expression
1		df-rmo 3364	. . 3 ⊢ (∃*𝑦 ∈ 𝐵 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))
2	1	ralbii 3083	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))
3		df-ral 3053	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)))
4		moanimv 2619	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)))
5	4	bicomi 224	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ ∃*𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)))
6		3anass 1094	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)))
7	6	bicomi 224	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑))
8	7	mobii 2548	. . . . 5 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑))
9	5, 8	bitri 275	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑))
10	9	albii 1819	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑))
11	3, 10	bitri 275	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑))
12	2, 11	bitri 275	1 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   ∈ wcel 2109  ∃*wmo 2538  ∀wral 3052  ∃*wrmo 3363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1780  df-mo 2540  df-ral 3053  df-rmo 3364

This theorem is referenced by:  2reu5lem3  3745