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Theorem intab 4442
 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 7040 or abexssex 7041 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 2614 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴))
21anbi2d 736 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝜑𝑧 = 𝐴) ↔ (𝜑𝑥 = 𝐴)))
32exbidv 1837 . . . . . . . 8 (𝑧 = 𝑥 → (∃𝑦(𝜑𝑧 = 𝐴) ↔ ∃𝑦(𝜑𝑥 = 𝐴)))
43cbvabv 2734 . . . . . . 7 {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} = {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)}
5 intab.2 . . . . . . 7 {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)} ∈ V
64, 5eqeltri 2684 . . . . . 6 {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ∈ V
7 nfe1 2014 . . . . . . . . 9 𝑦𝑦(𝜑𝑧 = 𝐴)
87nfab 2755 . . . . . . . 8 𝑦{𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}
98nfeq2 2766 . . . . . . 7 𝑦 𝑥 = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}
10 eleq2 2677 . . . . . . . 8 (𝑥 = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} → (𝐴𝑥𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}))
1110imbi2d 329 . . . . . . 7 (𝑥 = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} → ((𝜑𝐴𝑥) ↔ (𝜑𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})))
129, 11albid 2077 . . . . . 6 (𝑥 = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} → (∀𝑦(𝜑𝐴𝑥) ↔ ∀𝑦(𝜑𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})))
136, 12elab 3319 . . . . 5 ({𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ∈ {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} ↔ ∀𝑦(𝜑𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}))
14 19.8a 2039 . . . . . . . . 9 ((𝜑𝑧 = 𝐴) → ∃𝑦(𝜑𝑧 = 𝐴))
1514ex 449 . . . . . . . 8 (𝜑 → (𝑧 = 𝐴 → ∃𝑦(𝜑𝑧 = 𝐴)))
1615alrimiv 1842 . . . . . . 7 (𝜑 → ∀𝑧(𝑧 = 𝐴 → ∃𝑦(𝜑𝑧 = 𝐴)))
17 intab.1 . . . . . . . 8 𝐴 ∈ V
1817sbc6 3429 . . . . . . 7 ([𝐴 / 𝑧]𝑦(𝜑𝑧 = 𝐴) ↔ ∀𝑧(𝑧 = 𝐴 → ∃𝑦(𝜑𝑧 = 𝐴)))
1916, 18sylibr 223 . . . . . 6 (𝜑[𝐴 / 𝑧]𝑦(𝜑𝑧 = 𝐴))
20 df-sbc 3403 . . . . . 6 ([𝐴 / 𝑧]𝑦(𝜑𝑧 = 𝐴) ↔ 𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})
2119, 20sylib 207 . . . . 5 (𝜑𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)})
2213, 21mpgbir 1717 . . . 4 {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ∈ {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)}
23 intss1 4427 . . . 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 1790 . . . . . . . 8 ((∃𝑦(𝜑𝑧 = 𝐴) ∧ ∀𝑦(𝜑𝐴𝑥)) → ∃𝑦((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)))
26 simplr 788 . . . . . . . . . 10 (((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)) → 𝑧 = 𝐴)
27 pm3.35 609 . . . . . . . . . . 11 ((𝜑 ∧ (𝜑𝐴𝑥)) → 𝐴𝑥)
2827adantlr 747 . . . . . . . . . 10 (((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)) → 𝐴𝑥)
2926, 28eqeltrd 2688 . . . . . . . . 9 (((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)) → 𝑧𝑥)
3029exlimiv 1845 . . . . . . . 8 (∃𝑦((𝜑𝑧 = 𝐴) ∧ (𝜑𝐴𝑥)) → 𝑧𝑥)
3125, 30syl 17 . . . . . . 7 ((∃𝑦(𝜑𝑧 = 𝐴) ∧ ∀𝑦(𝜑𝐴𝑥)) → 𝑧𝑥)
3231ex 449 . . . . . 6 (∃𝑦(𝜑𝑧 = 𝐴) → (∀𝑦(𝜑𝐴𝑥) → 𝑧𝑥))
3332alrimiv 1842 . . . . 5 (∃𝑦(𝜑𝑧 = 𝐴) → ∀𝑥(∀𝑦(𝜑𝐴𝑥) → 𝑧𝑥))
34 vex 3176 . . . . . 6 𝑧 ∈ V
3534elintab 4422 . . . . 5 (𝑧 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} ↔ ∀𝑥(∀𝑦(𝜑𝐴𝑥) → 𝑧𝑥))
3633, 35sylibr 223 . . . 4 (∃𝑦(𝜑𝑧 = 𝐴) → 𝑧 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)})
3736abssi 3640 . . 3 {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)} ⊆ {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)}
3824, 37eqssi 3584 . 2 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} = {𝑧 ∣ ∃𝑦(𝜑𝑧 = 𝐴)}
3938, 4eqtri 2632 1 {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} = {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  Vcvv 3173  [wsbc 3402   ⊆ wss 3540  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-in 3547  df-ss 3554  df-int 4411 This theorem is referenced by: (None)
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