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Mirrors > Home > MPE Home > Th. List > r19.41vv | Structured version Visualization version GIF version |
Description: Version of r19.41v 3178 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 3178 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) | |
2 | 1 | rexbii 3083 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) |
3 | r19.41v 3178 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) | |
4 | 2, 3 | bitri 274 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∃wrex 3059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-rex 3060 |
This theorem is referenced by: genpass 11034 mulsuniflem 28099 addsdilem2 28102 mulsasslem1 28113 mulsasslem2 28114 dfcgra2 28706 axeuclid 28846 wspthsnwspthsnon 29799 dya2iocnrect 34032 satfv0 35099 satfv1 35104 satf0 35113 itg2addnclem3 37277 prprelprb 46994 prprspr2 46995 |
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