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Theorem intabs 4855
Description: Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)
Hypotheses
Ref Expression
intabs.1 (𝑥 = 𝑦 → (𝜑𝜓))
intabs.2 (𝑥 = {𝑦𝜓} → (𝜑𝜒))
intabs.3 ( {𝑦𝜓} ⊆ 𝐴𝜒)
Assertion
Ref Expression
intabs {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥𝜑}
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜑,𝑦   𝜓,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem intabs
StepHypRef Expression
1 sseq1 3659 . . . . . 6 (𝑥 = {𝑦𝜓} → (𝑥𝐴 {𝑦𝜓} ⊆ 𝐴))
2 intabs.2 . . . . . 6 (𝑥 = {𝑦𝜓} → (𝜑𝜒))
31, 2anbi12d 747 . . . . 5 (𝑥 = {𝑦𝜓} → ((𝑥𝐴𝜑) ↔ ( {𝑦𝜓} ⊆ 𝐴𝜒)))
4 intabs.3 . . . . 5 ( {𝑦𝜓} ⊆ 𝐴𝜒)
53, 4intmin3 4537 . . . 4 ( {𝑦𝜓} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
6 intnex 4851 . . . . 5 {𝑦𝜓} ∈ V ↔ {𝑦𝜓} = V)
7 ssv 3658 . . . . . 6 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ V
8 sseq2 3660 . . . . . 6 ( {𝑦𝜓} = V → ( {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓} ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ V))
97, 8mpbiri 248 . . . . 5 ( {𝑦𝜓} = V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
106, 9sylbi 207 . . . 4 {𝑦𝜓} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
115, 10pm2.61i 176 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓}
12 intabs.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
1312cbvabv 2776 . . . 4 {𝑥𝜑} = {𝑦𝜓}
1413inteqi 4511 . . 3 {𝑥𝜑} = {𝑦𝜓}
1511, 14sseqtr4i 3671 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑}
16 simpr 476 . . . 4 ((𝑥𝐴𝜑) → 𝜑)
1716ss2abi 3707 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑}
18 intss 4530 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑} → {𝑥𝜑} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
1917, 18ax-mp 5 . 2 {𝑥𝜑} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)}
2015, 19eqssi 3652 1 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231  wss 3607   cint 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-int 4508
This theorem is referenced by:  dfnn3  11072
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