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Theorem intab 3691
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). (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 2089 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴))
21anbi2d 452 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝜑𝑧 = 𝐴) ↔ (𝜑𝑥 = 𝐴)))
32exbidv 1748 . . . . . . . 8 (𝑧 = 𝑥 → (∃𝑦(𝜑𝑧 = 𝐴) ↔ ∃𝑦(𝜑𝑥 = 𝐴)))
43cbvabv 2206 . . . . . . 7 {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} = {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)}
5 intab.2 . . . . . . 7 {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)} ∈ V
64, 5eqeltri 2155 . . . . . 6 {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ∈ V
7 nfe1 1426 . . . . . . . . 9 𝑦𝑦(𝜑𝑧 = 𝐴)
87nfab 2227 . . . . . . . 8 𝑦{𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}
98nfeq2 2234 . . . . . . 7 𝑦 𝑥 = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}
10 eleq2 2146 . . . . . . . 8 (𝑥 = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} → (𝐴𝑥𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}))
1110imbi2d 228 . . . . . . 7 (𝑥 = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} → ((𝜑𝐴𝑥) ↔ (𝜑𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})))
129, 11albid 1547 . . . . . 6 (𝑥 = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} → (∀𝑦(𝜑𝐴𝑥) ↔ ∀𝑦(𝜑𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})))
136, 12elab 2748 . . . . 5 ({𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ∈ {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} ↔ ∀𝑦(𝜑𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}))
14 19.8a 1523 . . . . . . . . 9 ((𝜑𝑧 = 𝐴) → ∃𝑦(𝜑𝑧 = 𝐴))
1514ex 113 . . . . . . . 8 (𝜑 → (𝑧 = 𝐴 → ∃𝑦(𝜑𝑧 = 𝐴)))
1615alrimiv 1797 . . . . . . 7 (𝜑 → ∀𝑧(𝑧 = 𝐴 → ∃𝑦(𝜑𝑧 = 𝐴)))
17 intab.1 . . . . . . . 8 𝐴 ∈ V
1817sbc6 2851 . . . . . . 7 ([𝐴 / 𝑧]𝑦(𝜑𝑧 = 𝐴) ↔ ∀𝑧(𝑧 = 𝐴 → ∃𝑦(𝜑𝑧 = 𝐴)))
1916, 18sylibr 132 . . . . . 6 (𝜑[𝐴 / 𝑧]𝑦(𝜑𝑧 = 𝐴))
20 df-sbc 2827 . . . . . 6 ([𝐴 / 𝑧]𝑦(𝜑𝑧 = 𝐴) ↔ 𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})
2119, 20sylib 120 . . . . 5 (𝜑𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})
2213, 21mpgbir 1383 . . . 4 {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ∈ {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)}
23 intss1 3677 . . . 4 ({𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ∈ {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} → {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} ⊆ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})
2422, 23ax-mp 7 . . 3 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} ⊆ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}
25 19.29r 1553 . . . . . . . 8 ((∃𝑦(𝜑𝑧 = 𝐴) ∧ ∀𝑦(𝜑𝐴𝑥)) → ∃𝑦((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)))
26 simplr 497 . . . . . . . . . 10 (((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)) → 𝑧 = 𝐴)
27 pm3.35 339 . . . . . . . . . . 11 ((𝜑 ∧ (𝜑𝐴𝑥)) → 𝐴𝑥)
2827adantlr 461 . . . . . . . . . 10 (((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)) → 𝐴𝑥)
2926, 28eqeltrd 2159 . . . . . . . . 9 (((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)) → 𝑧𝑥)
3029exlimiv 1530 . . . . . . . 8 (∃𝑦((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)) → 𝑧𝑥)
3125, 30syl 14 . . . . . . 7 ((∃𝑦(𝜑𝑧 = 𝐴) ∧ ∀𝑦(𝜑𝐴𝑥)) → 𝑧𝑥)
3231ex 113 . . . . . 6 (∃𝑦(𝜑𝑧 = 𝐴) → (∀𝑦(𝜑𝐴𝑥) → 𝑧𝑥))
3332alrimiv 1797 . . . . 5 (∃𝑦(𝜑𝑧 = 𝐴) → ∀𝑥(∀𝑦(𝜑𝐴𝑥) → 𝑧𝑥))
34 vex 2615 . . . . . 6 𝑧 ∈ V
3534elintab 3673 . . . . 5 (𝑧 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} ↔ ∀𝑥(∀𝑦(𝜑𝐴𝑥) → 𝑧𝑥))
3633, 35sylibr 132 . . . 4 (∃𝑦(𝜑𝑧 = 𝐴) → 𝑧 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)})
3736abssi 3080 . . 3 {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ⊆ {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)}
3824, 37eqssi 3026 . 2 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}
3938, 4eqtri 2103 1 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} = {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283   = wceq 1285  wex 1422  wcel 1434  {cab 2069  Vcvv 2612  [wsbc 2826  wss 2984   cint 3662
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-sbc 2827  df-in 2990  df-ss 2997  df-int 3663
This theorem is referenced by: (None)
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