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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 1551 | . . 3 ⊢ (∃y∃z∃w(φ ∧ ψ) ↔ ∃z∃y∃w(φ ∧ ψ)) | |
2 | 1 | exbii 1493 | . 2 ⊢ (∃x∃y∃z∃w(φ ∧ ψ) ↔ ∃x∃z∃y∃w(φ ∧ ψ)) |
3 | eeanv 1804 | . . 3 ⊢ (∃y∃w(φ ∧ ψ) ↔ (∃yφ ∧ ∃wψ)) | |
4 | 3 | 2exbii 1494 | . 2 ⊢ (∃x∃z∃y∃w(φ ∧ ψ) ↔ ∃x∃z(∃yφ ∧ ∃wψ)) |
5 | eeanv 1804 | . 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 1378 |
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 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-nf 1347 |
This theorem is referenced by: ee8anv 1807 cgsex4g 2585 th3qlem1 6144 dmaddpq 6363 dmmulpq 6364 ltdcnq 6381 enq0ref 6416 nqpnq0nq 6436 nqnq0a 6437 nqnq0m 6438 genpdisj 6506 axaddcl 6750 axmulcl 6752 |
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