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Mirrors > Home > MPE Home > Th. List > 2reu5lem1 | Structured version Visualization version GIF version |
Description: Lemma for 2reu5 3754. Note that ∃!𝑥 ∈ 𝐴∃!𝑦 ∈ 𝐵𝜑 does not mean "there is exactly one 𝑥 in 𝐴 and exactly one 𝑦 in 𝐵 such that 𝜑 holds"; see comment for 2eu5 2651. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Ref | Expression |
---|---|
2reu5lem1 | ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑥∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 3377 | . . 3 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) | |
2 | 1 | reubii 3385 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) |
3 | df-reu 3377 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))) | |
4 | euanv 2620 | . . . . . 6 ⊢ (∃!𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))) | |
5 | 4 | bicomi 223 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ ∃!𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑))) |
6 | 3anass 1095 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑))) | |
7 | 6 | bicomi 223 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) |
8 | 7 | eubii 2579 | . . . . 5 ⊢ (∃!𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ ∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) |
9 | 5, 8 | bitri 274 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ ∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) |
10 | 9 | eubii 2579 | . . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ ∃!𝑥∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) |
11 | 3, 10 | bitri 274 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∃!𝑥∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) |
12 | 2, 11 | bitri 274 | 1 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑥∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 ∃!weu 2562 ∃!wreu 3374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-ex 1782 df-mo 2534 df-eu 2563 df-reu 3377 |
This theorem is referenced by: 2reu5lem3 3753 |
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