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Theorem cbvrex2v 2592
Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
Hypotheses
Ref Expression
cbvrex2v.1 (𝑥 = 𝑧 → (𝜑𝜒))
cbvrex2v.2 (𝑦 = 𝑤 → (𝜒𝜓))
Assertion
Ref Expression
cbvrex2v (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑧,𝐴   𝑤,𝐵   𝑥,𝐵,𝑦   𝑧,𝐵,𝑦   𝜒,𝑤   𝜒,𝑥   𝜑,𝑧   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑤)   𝜓(𝑥,𝑧,𝑤)   𝜒(𝑦,𝑧)   𝐴(𝑦,𝑤)

Proof of Theorem cbvrex2v
StepHypRef Expression
1 cbvrex2v.1 . . . 4 (𝑥 = 𝑧 → (𝜑𝜒))
21rexbidv 2375 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜒))
32cbvrexv 2584 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑦𝐵 𝜒)
4 cbvrex2v.2 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
54cbvrexv 2584 . . 3 (∃𝑦𝐵 𝜒 ↔ ∃𝑤𝐵 𝜓)
65rexbii 2379 . 2 (∃𝑧𝐴𝑦𝐵 𝜒 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
73, 6bitri 182 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359
This theorem is referenced by:  eroveu  6313  genipv  6971  bezoutlemnewy  10765
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