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Theorem cbvopab1 4007
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
cbvopab1.1 𝑧𝜑
cbvopab1.2 𝑥𝜓
cbvopab1.3 (𝑥 = 𝑧 → (𝜑𝜓))
Assertion
Ref Expression
cbvopab1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Distinct variable groups:   𝑥,𝑦   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem cbvopab1
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1509 . . . . 5 𝑣𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
2 nfv 1509 . . . . . . 7 𝑥 𝑤 = ⟨𝑣, 𝑦
3 nfs1v 1913 . . . . . . 7 𝑥[𝑣 / 𝑥]𝜑
42, 3nfan 1545 . . . . . 6 𝑥(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
54nfex 1617 . . . . 5 𝑥𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
6 opeq1 3711 . . . . . . . 8 (𝑥 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑦⟩)
76eqeq2d 2152 . . . . . . 7 (𝑥 = 𝑣 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑣, 𝑦⟩))
8 sbequ12 1745 . . . . . . 7 (𝑥 = 𝑣 → (𝜑 ↔ [𝑣 / 𝑥]𝜑))
97, 8anbi12d 465 . . . . . 6 (𝑥 = 𝑣 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)))
109exbidv 1798 . . . . 5 (𝑥 = 𝑣 → (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)))
111, 5, 10cbvex 1730 . . . 4 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑣𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑))
12 nfv 1509 . . . . . . 7 𝑧 𝑤 = ⟨𝑣, 𝑦
13 cbvopab1.1 . . . . . . . 8 𝑧𝜑
1413nfsb 1920 . . . . . . 7 𝑧[𝑣 / 𝑥]𝜑
1512, 14nfan 1545 . . . . . 6 𝑧(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
1615nfex 1617 . . . . 5 𝑧𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
17 nfv 1509 . . . . 5 𝑣𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)
18 opeq1 3711 . . . . . . . 8 (𝑣 = 𝑧 → ⟨𝑣, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
1918eqeq2d 2152 . . . . . . 7 (𝑣 = 𝑧 → (𝑤 = ⟨𝑣, 𝑦⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
20 sbequ 1813 . . . . . . . 8 (𝑣 = 𝑧 → ([𝑣 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
21 cbvopab1.2 . . . . . . . . 9 𝑥𝜓
22 cbvopab1.3 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜑𝜓))
2321, 22sbie 1765 . . . . . . . 8 ([𝑧 / 𝑥]𝜑𝜓)
2420, 23syl6bb 195 . . . . . . 7 (𝑣 = 𝑧 → ([𝑣 / 𝑥]𝜑𝜓))
2519, 24anbi12d 465 . . . . . 6 (𝑣 = 𝑧 → ((𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)))
2625exbidv 1798 . . . . 5 (𝑣 = 𝑧 → (∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)))
2716, 17, 26cbvex 1730 . . . 4 (∃𝑣𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓))
2811, 27bitri 183 . . 3 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓))
2928abbii 2256 . 2 {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)}
30 df-opab 3996 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31 df-opab 3996 . 2 {⟨𝑧, 𝑦⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)}
3229, 30, 313eqtr4i 2171 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wnf 1437  wex 1469  [wsb 1736  {cab 2126  cop 3533  {copab 3994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3078  df-sn 3536  df-pr 3537  df-op 3539  df-opab 3996
This theorem is referenced by:  cbvopab1v  4010  cbvmptf  4028  cbvmpt  4029
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