ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cbvex2 GIF version

Theorem cbvex2 1797
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
cbval2.1 𝑧𝜑
cbval2.2 𝑤𝜑
cbval2.3 𝑥𝜓
cbval2.4 𝑦𝜓
cbval2.5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbvex2 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
Distinct variable groups:   𝑥,𝑦   𝑦,𝑧   𝑥,𝑤   𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvex2
StepHypRef Expression
1 cbval2.1 . . 3 𝑧𝜑
21nfex 1528 . 2 𝑧𝑦𝜑
3 cbval2.3 . . 3 𝑥𝜓
43nfex 1528 . 2 𝑥𝑤𝜓
5 nfv 1421 . . . . . 6 𝑤 𝑥 = 𝑧
6 cbval2.2 . . . . . 6 𝑤𝜑
75, 6nfan 1457 . . . . 5 𝑤(𝑥 = 𝑧𝜑)
8 nfv 1421 . . . . . 6 𝑦 𝑥 = 𝑧
9 cbval2.4 . . . . . 6 𝑦𝜓
108, 9nfan 1457 . . . . 5 𝑦(𝑥 = 𝑧𝜓)
11 cbval2.5 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
1211expcom 109 . . . . . 6 (𝑦 = 𝑤 → (𝑥 = 𝑧 → (𝜑𝜓)))
1312pm5.32d 423 . . . . 5 (𝑦 = 𝑤 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧𝜓)))
147, 10, 13cbvex 1639 . . . 4 (∃𝑦(𝑥 = 𝑧𝜑) ↔ ∃𝑤(𝑥 = 𝑧𝜓))
15 19.42v 1786 . . . 4 (∃𝑦(𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧 ∧ ∃𝑦𝜑))
16 19.42v 1786 . . . 4 (∃𝑤(𝑥 = 𝑧𝜓) ↔ (𝑥 = 𝑧 ∧ ∃𝑤𝜓))
1714, 15, 163bitr3i 199 . . 3 ((𝑥 = 𝑧 ∧ ∃𝑦𝜑) ↔ (𝑥 = 𝑧 ∧ ∃𝑤𝜓))
18 pm5.32 426 . . 3 ((𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓)) ↔ ((𝑥 = 𝑧 ∧ ∃𝑦𝜑) ↔ (𝑥 = 𝑧 ∧ ∃𝑤𝜓)))
1917, 18mpbir 134 . 2 (𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓))
202, 4, 19cbvex 1639 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wnf 1349  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-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-nf 1350
This theorem is referenced by:  cbvex2v  1799  cbvopab  3828  cbvoprab12  5578
  Copyright terms: Public domain W3C validator