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Theorem bdzfauscl 10369
Description: Closed form of the version of zfauscl 3904 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 2117 . . . . . 6 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
21anbi1d 446 . . . . 5 (𝑧 = 𝐴 → ((𝑥𝑧𝜑) ↔ (𝑥𝐴𝜑)))
32bibi2d 225 . . . 4 (𝑧 = 𝐴 → ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝐴𝜑))))
43albidv 1721 . . 3 (𝑧 = 𝐴 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
54exbidv 1722 . 2 (𝑧 = 𝐴 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
6 bdzfauscl.bd . . 3 BOUNDED 𝜑
76bdsep1 10364 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
85, 7vtoclg 2630 1 (𝐴𝑉 → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  BOUNDED wbd 10291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-bdsep 10363
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  bdinex1  10378
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