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Theorem intab 4477
Description: The intersection of a special case of a class abstraction. 𝑦 may be free in 𝜑 and 𝐴, which can be thought of a 𝜑(𝑦) and 𝐴(𝑦). Typically, abrexex2 7097 or abexssex 7098 can be used to satisfy the second hypothesis. (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
intab.1 𝐴 ∈ V
intab.2 {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)} ∈ V
Assertion
Ref Expression
intab {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} = {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)}
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem intab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2630 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴))
21anbi2d 739 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝜑𝑧 = 𝐴) ↔ (𝜑𝑥 = 𝐴)))
32exbidv 1852 . . . . . . . 8 (𝑧 = 𝑥 → (∃𝑦(𝜑𝑧 = 𝐴) ↔ ∃𝑦(𝜑𝑥 = 𝐴)))
43cbvabv 2750 . . . . . . 7 {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} = {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)}
5 intab.2 . . . . . . 7 {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)} ∈ V
64, 5eqeltri 2700 . . . . . 6 {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ∈ V
7 nfe1 2029 . . . . . . . . 9 𝑦𝑦(𝜑𝑧 = 𝐴)
87nfab 2771 . . . . . . . 8 𝑦{𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}
98nfeq2 2782 . . . . . . 7 𝑦 𝑥 = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}
10 eleq2 2693 . . . . . . . 8 (𝑥 = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} → (𝐴𝑥𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}))
1110imbi2d 330 . . . . . . 7 (𝑥 = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} → ((𝜑𝐴𝑥) ↔ (𝜑𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})))
129, 11albid 2093 . . . . . 6 (𝑥 = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} → (∀𝑦(𝜑𝐴𝑥) ↔ ∀𝑦(𝜑𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})))
136, 12elab 3338 . . . . 5 ({𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ∈ {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} ↔ ∀𝑦(𝜑𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}))
14 19.8a 2054 . . . . . . . . 9 ((𝜑𝑧 = 𝐴) → ∃𝑦(𝜑𝑧 = 𝐴))
1514ex 450 . . . . . . . 8 (𝜑 → (𝑧 = 𝐴 → ∃𝑦(𝜑𝑧 = 𝐴)))
1615alrimiv 1857 . . . . . . 7 (𝜑 → ∀𝑧(𝑧 = 𝐴 → ∃𝑦(𝜑𝑧 = 𝐴)))
17 intab.1 . . . . . . . 8 𝐴 ∈ V
1817sbc6 3449 . . . . . . 7 ([𝐴 / 𝑧]𝑦(𝜑𝑧 = 𝐴) ↔ ∀𝑧(𝑧 = 𝐴 → ∃𝑦(𝜑𝑧 = 𝐴)))
1916, 18sylibr 224 . . . . . 6 (𝜑[𝐴 / 𝑧]𝑦(𝜑𝑧 = 𝐴))
20 df-sbc 3423 . . . . . 6 ([𝐴 / 𝑧]𝑦(𝜑𝑧 = 𝐴) ↔ 𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})
2119, 20sylib 208 . . . . 5 (𝜑𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})
2213, 21mpgbir 1723 . . . 4 {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ∈ {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)}
23 intss1 4462 . . . 4 ({𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ∈ {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} → {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} ⊆ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})
2422, 23ax-mp 5 . . 3 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} ⊆ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}
25 19.29r 1801 . . . . . . . 8 ((∃𝑦(𝜑𝑧 = 𝐴) ∧ ∀𝑦(𝜑𝐴𝑥)) → ∃𝑦((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)))
26 simplr 791 . . . . . . . . . 10 (((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)) → 𝑧 = 𝐴)
27 pm3.35 610 . . . . . . . . . . 11 ((𝜑 ∧ (𝜑𝐴𝑥)) → 𝐴𝑥)
2827adantlr 750 . . . . . . . . . 10 (((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)) → 𝐴𝑥)
2926, 28eqeltrd 2704 . . . . . . . . 9 (((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)) → 𝑧𝑥)
3029exlimiv 1860 . . . . . . . 8 (∃𝑦((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)) → 𝑧𝑥)
3125, 30syl 17 . . . . . . 7 ((∃𝑦(𝜑𝑧 = 𝐴) ∧ ∀𝑦(𝜑𝐴𝑥)) → 𝑧𝑥)
3231ex 450 . . . . . 6 (∃𝑦(𝜑𝑧 = 𝐴) → (∀𝑦(𝜑𝐴𝑥) → 𝑧𝑥))
3332alrimiv 1857 . . . . 5 (∃𝑦(𝜑𝑧 = 𝐴) → ∀𝑥(∀𝑦(𝜑𝐴𝑥) → 𝑧𝑥))
34 vex 3194 . . . . . 6 𝑧 ∈ V
3534elintab 4457 . . . . 5 (𝑧 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} ↔ ∀𝑥(∀𝑦(𝜑𝐴𝑥) → 𝑧𝑥))
3633, 35sylibr 224 . . . 4 (∃𝑦(𝜑𝑧 = 𝐴) → 𝑧 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)})
3736abssi 3661 . . 3 {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ⊆ {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)}
3824, 37eqssi 3604 . 2 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}
3938, 4eqtri 2648 1 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} = {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1992  {cab 2612  Vcvv 3191  [wsbc 3422  wss 3560   cint 4445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-sbc 3423  df-in 3567  df-ss 3574  df-int 4446
This theorem is referenced by: (None)
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