Theorem bdzfauscl 10681

Description: Closed form of the version of zfauscl 3898 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)

Hypothesis

Ref	Expression
bdzfauscl.bd	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdzfauscl	⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))

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.

Step	Hyp	Ref	Expression
1		eleq2 2142	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐴))
2	1	anbi1d 452	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑥 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
3	2	bibi2d 230	. . . 4 ⊢ (𝑧 = 𝐴 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))))
4	3	albidv 1745	. . 3 ⊢ (𝑧 = 𝐴 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))))
5	4	exbidv 1746	. 2 ⊢ (𝑧 = 𝐴 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))))
6		bdzfauscl.bd	. . 3 ⊢ BOUNDED 𝜑
7	6	bdsep1 10676	. 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
8	5, 7	vtoclg 2658	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282   = wceq 1284  ∃wex 1421   ∈ wcel 1433  BOUNDED wbd 10603

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  bdinex1  10690