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Theorem bdzfauscl 13232
Description: Closed form of the version of zfauscl 4051 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
Hypothesis
Ref Expression
bdzfauscl.bd BOUNDED 𝜑
Assertion
Ref Expression
bdzfauscl (𝐴𝑉 → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem bdzfauscl
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2203 . . . . . 6 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
21anbi1d 460 . . . . 5 (𝑧 = 𝐴 → ((𝑥𝑧𝜑) ↔ (𝑥𝐴𝜑)))
32bibi2d 231 . . . 4 (𝑧 = 𝐴 → ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝐴𝜑))))
43albidv 1796 . . 3 (𝑧 = 𝐴 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
54exbidv 1797 . 2 (𝑧 = 𝐴 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
6 bdzfauscl.bd . . 3 BOUNDED 𝜑
76bdsep1 13227 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
85, 7vtoclg 2746 1 (𝐴𝑉 → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1329   = wceq 1331  wex 1468  wcel 1480  BOUNDED wbd 13154
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bdsep 13226
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by:  bdinex1  13241
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