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Mirrors > Home > MPE Home > Th. List > 19.41vv | Structured version Visualization version GIF version |
Description: Version of 19.41 2090 with two quantifiers and a dv 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 1901 | . . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ 𝜓)) | |
2 | 1 | exbii 1764 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜓)) |
3 | 19.41v 1901 | . 2 ⊢ (∃𝑥(∃𝑦𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓)) | |
4 | 2, 3 | bitri 263 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: 19.41vvv 1903 rabxp 5078 copsex2gb 5153 mpt2mptx 6649 xpassen 7939 dfac5lem1 8829 3v3e3cycl 26193 bnj996 30279 dfdm5 30921 dfrn5 30922 elima4 30924 brtxp2 31158 brpprod3a 31163 brimg 31214 brsuccf 31218 diblsmopel 35478 fusgr2wsp2nb 41498 mpt2mptx2 41906 |
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