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Theorem rexcom13 2492
Description: Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
Assertion
Ref Expression
rexcom13 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑧,𝐵   𝑥,𝑦,𝐶
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑧)

Proof of Theorem rexcom13
StepHypRef Expression
1 rexcom 2491 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑦𝐵𝑥𝐴𝑧𝐶 𝜑)
2 rexcom 2491 . . 3 (∃𝑥𝐴𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑥𝐴 𝜑)
32rexbii 2348 . 2 (∃𝑦𝐵𝑥𝐴𝑧𝐶 𝜑 ↔ ∃𝑦𝐵𝑧𝐶𝑥𝐴 𝜑)
4 rexcom 2491 . 2 (∃𝑦𝐵𝑧𝐶𝑥𝐴 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
51, 3, 43bitri 199 1 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wb 102  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329
This theorem is referenced by:  rexrot4  2493
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