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Mirrors > Home > MPE Home > Th. List > 19.41vv | Structured version Visualization version GIF version |
Description: Version of 19.41 2233 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.) |
Ref | Expression |
---|---|
19.41vv | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.41v 1947 | . . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ 𝜓)) | |
2 | 1 | exbii 1845 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜓)) |
3 | 19.41v 1947 | . 2 ⊢ (∃𝑥(∃𝑦𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓)) | |
4 | 2, 3 | bitri 275 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 |
This theorem is referenced by: 19.41vvv 1949 cgsex4g 3526 rabxp 5737 copsex2gb 5819 mpomptx 7546 xpassen 9105 dfac5lem1 10161 fusgr2wsp2nb 30363 bnj996 34949 dfdm5 35754 dfrn5 35755 elima4 35757 brtxp2 35863 brpprod3a 35868 brimg 35919 brsuccf 35923 brxrn2 38357 diblsmopel 41154 en2pr 43537 mpomptx2 48180 |
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