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Mirrors > Home > ILE Home > Th. List > ee4anv | GIF version |
Description: Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.) |
Ref | Expression |
---|---|
ee4anv | ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom 1554 | . . 3 ⊢ (∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓)) | |
2 | 1 | exbii 1496 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓)) |
3 | eeanv 1807 | . . 3 ⊢ (∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ ∃𝑤𝜓)) | |
4 | 3 | 2exbii 1497 | . 2 ⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓)) |
5 | eeanv 1807 | . 2 ⊢ (∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) | |
6 | 2, 4, 5 | 3bitri 195 | 1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: ee8anv 1810 cgsex4g 2591 th3qlem1 6208 dmaddpq 6477 dmmulpq 6478 ltdcnq 6495 enq0ref 6531 nqpnq0nq 6551 nqnq0a 6552 nqnq0m 6553 genpdisj 6621 axaddcl 6940 axmulcl 6942 |
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