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Mirrors > Home > MPE Home > Th. List > r19.41vv | Structured version Visualization version GIF version |
Description: Version of r19.41v 3250 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 3250 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) | |
2 | 1 | rexbii 3160 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) |
3 | r19.41v 3250 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) | |
4 | 2, 3 | bitri 278 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wrex 3052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-rex 3057 |
This theorem is referenced by: genpass 10588 dfcgra2 26875 axeuclid 27008 wspthsnwspthsnon 27954 dya2iocnrect 31914 satfv0 32987 satfv1 32992 satf0 33001 itg2addnclem3 35516 prprelprb 44585 prprspr2 44586 |
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