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Theorem cbvrexf 2573
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 1462 . . . 4 𝑧(𝑥𝐴𝜑)
2 cbvralf.1 . . . . . 6 𝑥𝐴
32nfcri 2214 . . . . 5 𝑥 𝑧𝐴
4 nfs1v 1857 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfan 1498 . . . 4 𝑥(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
6 eleq1 2142 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
7 sbequ12 1695 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
86, 7anbi12d 457 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)))
91, 5, 8cbvex 1680 . . 3 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
10 cbvralf.2 . . . . . 6 𝑦𝐴
1110nfcri 2214 . . . . 5 𝑦 𝑧𝐴
12 cbvralf.3 . . . . . 6 𝑦𝜑
1312nfsb 1864 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1411, 13nfan 1498 . . . 4 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
15 nfv 1462 . . . 4 𝑧(𝑦𝐴𝜓)
16 eleq1 2142 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
17 sbequ 1762 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18 cbvralf.4 . . . . . . 7 𝑥𝜓
19 cbvralf.5 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
2018, 19sbie 1715 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
2117, 20syl6bb 194 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
2216, 21anbi12d 457 . . . 4 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2314, 15, 22cbvex 1680 . . 3 (∃𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃𝑦(𝑦𝐴𝜓))
249, 23bitri 182 . 2 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑦(𝑦𝐴𝜓))
25 df-rex 2355 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
26 df-rex 2355 . 2 (∃𝑦𝐴 𝜓 ↔ ∃𝑦(𝑦𝐴𝜓))
2724, 25, 263bitr4i 210 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wnf 1390  wex 1422  wcel 1434  [wsb 1686  wnfc 2207  wrex 2350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355
This theorem is referenced by:  cbvrex  2575
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