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Theorem cbvopab 4086
Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
Hypotheses
Ref Expression
cbvopab.1 𝑧𝜑
cbvopab.2 𝑤𝜑
cbvopab.3 𝑥𝜓
cbvopab.4 𝑦𝜓
cbvopab.5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbvopab {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvopab
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nfv 1538 . . . . 5 𝑧 𝑣 = ⟨𝑥, 𝑦
2 cbvopab.1 . . . . 5 𝑧𝜑
31, 2nfan 1575 . . . 4 𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4 nfv 1538 . . . . 5 𝑤 𝑣 = ⟨𝑥, 𝑦
5 cbvopab.2 . . . . 5 𝑤𝜑
64, 5nfan 1575 . . . 4 𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7 nfv 1538 . . . . 5 𝑥 𝑣 = ⟨𝑧, 𝑤
8 cbvopab.3 . . . . 5 𝑥𝜓
97, 8nfan 1575 . . . 4 𝑥(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)
10 nfv 1538 . . . . 5 𝑦 𝑣 = ⟨𝑧, 𝑤
11 cbvopab.4 . . . . 5 𝑦𝜓
1210, 11nfan 1575 . . . 4 𝑦(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)
13 opeq12 3792 . . . . . 6 ((𝑥 = 𝑧𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
1413eqeq2d 2199 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑧, 𝑤⟩))
15 cbvopab.5 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
1614, 15anbi12d 473 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)))
173, 6, 9, 12, 16cbvex2 1932 . . 3 (∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓))
1817abbii 2303 . 2 {𝑣 ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑧𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
19 df-opab 4077 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20 df-opab 4077 . 2 {⟨𝑧, 𝑤⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑧𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
2118, 19, 203eqtr4i 2218 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wnf 1470  wex 1502  {cab 2173  cop 3607  {copab 4075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-opab 4077
This theorem is referenced by:  cbvopabv  4087  opelopabsb  4272
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