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Theorem cbvrexf 2650
Description: Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
Hypotheses
Ref Expression
cbvralf.1 𝑥𝐴
cbvralf.2 𝑦𝐴
cbvralf.3 𝑦𝜑
cbvralf.4 𝑥𝜓
cbvralf.5 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrexf (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)

Proof of Theorem cbvrexf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1509 . . . 4 𝑧(𝑥𝐴𝜑)
2 cbvralf.1 . . . . . 6 𝑥𝐴
32nfcri 2276 . . . . 5 𝑥 𝑧𝐴
4 nfs1v 1913 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfan 1545 . . . 4 𝑥(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
6 eleq1 2203 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
7 sbequ12 1745 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
86, 7anbi12d 465 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)))
91, 5, 8cbvex 1730 . . 3 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
10 cbvralf.2 . . . . . 6 𝑦𝐴
1110nfcri 2276 . . . . 5 𝑦 𝑧𝐴
12 cbvralf.3 . . . . . 6 𝑦𝜑
1312nfsb 1920 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1411, 13nfan 1545 . . . 4 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
15 nfv 1509 . . . 4 𝑧(𝑦𝐴𝜓)
16 eleq1 2203 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
17 sbequ 1813 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18 cbvralf.4 . . . . . . 7 𝑥𝜓
19 cbvralf.5 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
2018, 19sbie 1765 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
2117, 20syl6bb 195 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
2216, 21anbi12d 465 . . . 4 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2314, 15, 22cbvex 1730 . . 3 (∃𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃𝑦(𝑦𝐴𝜓))
249, 23bitri 183 . 2 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑦(𝑦𝐴𝜓))
25 df-rex 2423 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
26 df-rex 2423 . 2 (∃𝑦𝐴 𝜓 ↔ ∃𝑦(𝑦𝐴𝜓))
2724, 25, 263bitr4i 211 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wnf 1437  wex 1469  wcel 1481  [wsb 1736  wnfc 2269  wrex 2418
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-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423
This theorem is referenced by:  cbvrex  2652
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