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Theorem cbvex2 1813
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 1544 . 2 𝑧𝑦𝜑
3 cbval2.3 . . 3 𝑥𝜓
43nfex 1544 . 2 𝑥𝑤𝜓
5 nfv 1437 . . . . . 6 𝑤 𝑥 = 𝑧
6 cbval2.2 . . . . . 6 𝑤𝜑
75, 6nfan 1473 . . . . 5 𝑤(𝑥 = 𝑧𝜑)
8 nfv 1437 . . . . . 6 𝑦 𝑥 = 𝑧
9 cbval2.4 . . . . . 6 𝑦𝜓
108, 9nfan 1473 . . . . 5 𝑦(𝑥 = 𝑧𝜓)
11 cbval2.5 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
1211expcom 113 . . . . . 6 (𝑦 = 𝑤 → (𝑥 = 𝑧 → (𝜑𝜓)))
1312pm5.32d 431 . . . . 5 (𝑦 = 𝑤 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧𝜓)))
147, 10, 13cbvex 1655 . . . 4 (∃𝑦(𝑥 = 𝑧𝜑) ↔ ∃𝑤(𝑥 = 𝑧𝜓))
15 19.42v 1802 . . . 4 (∃𝑦(𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧 ∧ ∃𝑦𝜑))
16 19.42v 1802 . . . 4 (∃𝑤(𝑥 = 𝑧𝜓) ↔ (𝑥 = 𝑧 ∧ ∃𝑤𝜓))
1714, 15, 163bitr3i 203 . . 3 ((𝑥 = 𝑧 ∧ ∃𝑦𝜑) ↔ (𝑥 = 𝑧 ∧ ∃𝑤𝜓))
18 pm5.32 434 . . 3 ((𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓)) ↔ ((𝑥 = 𝑧 ∧ ∃𝑦𝜑) ↔ (𝑥 = 𝑧 ∧ ∃𝑤𝜓)))
1917, 18mpbir 138 . 2 (𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓))
202, 4, 19cbvex 1655 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wnf 1365  wex 1397
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-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  cbvex2v  1815  cbvopab  3856  cbvoprab12  5606
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