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Theorem bdzfauscl 14264
Description: Closed form of the version of zfauscl 4120 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 2241 . . . . . 6 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
21anbi1d 465 . . . . 5 (𝑧 = 𝐴 → ((𝑥𝑧𝜑) ↔ (𝑥𝐴𝜑)))
32bibi2d 232 . . . 4 (𝑧 = 𝐴 → ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝐴𝜑))))
43albidv 1824 . . 3 (𝑧 = 𝐴 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
54exbidv 1825 . 2 (𝑧 = 𝐴 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
6 bdzfauscl.bd . . 3 BOUNDED 𝜑
76bdsep1 14259 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
85, 7vtoclg 2797 1 (𝐴𝑉 → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  wex 1492  wcel 2148  BOUNDED wbd 14186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bdsep 14258
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  bdinex1  14273
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