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| Mirrors > Home > MPE Home > Th. List > r19.41vv | Structured version Visualization version GIF version | ||
| Description: Version of r19.41v 3189 with two quantifiers. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
| Ref | Expression |
|---|---|
| r19.41vv | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r19.41v 3189 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) | |
| 2 | 1 | rexbii 3094 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) |
| 3 | r19.41v 3189 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) | |
| 4 | 2, 3 | bitri 275 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∃wrex 3070 |
| 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-rex 3071 |
| This theorem is referenced by: genpass 11049 mulsuniflem 28175 addsdilem2 28178 mulsasslem1 28189 mulsasslem2 28190 dfcgra2 28838 axeuclid 28978 wspthsnwspthsnon 29936 dya2iocnrect 34283 satfv0 35363 satfv1 35368 satf0 35377 itg2addnclem3 37680 prprelprb 47504 prprspr2 47505 |
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