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Theorem intabs 5335
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 4003 . . . . . 6 (𝑥 = {𝑦𝜓} → (𝑥𝐴 {𝑦𝜓} ⊆ 𝐴))
2 intabs.2 . . . . . 6 (𝑥 = {𝑦𝜓} → (𝜑𝜒))
31, 2anbi12d 631 . . . . 5 (𝑥 = {𝑦𝜓} → ((𝑥𝐴𝜑) ↔ ( {𝑦𝜓} ⊆ 𝐴𝜒)))
4 intabs.3 . . . . 5 ( {𝑦𝜓} ⊆ 𝐴𝜒)
53, 4intmin3 4973 . . . 4 ( {𝑦𝜓} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
6 intnex 5331 . . . . 5 {𝑦𝜓} ∈ V ↔ {𝑦𝜓} = V)
7 ssv 4002 . . . . . 6 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ V
8 sseq2 4004 . . . . . 6 ( {𝑦𝜓} = V → ( {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓} ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ V))
97, 8mpbiri 257 . . . . 5 ( {𝑦𝜓} = V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
106, 9sylbi 216 . . . 4 {𝑦𝜓} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
115, 10pm2.61i 182 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓}
12 intabs.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
1312cbvabv 2804 . . . 4 {𝑥𝜑} = {𝑦𝜓}
1413inteqi 4947 . . 3 {𝑥𝜑} = {𝑦𝜓}
1511, 14sseqtrri 4015 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑}
16 simpr 485 . . . 4 ((𝑥𝐴𝜑) → 𝜑)
1716ss2abi 4059 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑}
18 intss 4966 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑} → {𝑥𝜑} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
1917, 18ax-mp 5 . 2 {𝑥𝜑} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)}
2015, 19eqssi 3994 1 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2708  Vcvv 3473  wss 3944   cint 4943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-in 3951  df-ss 3961  df-nul 4319  df-int 4944
This theorem is referenced by:  dfnn3  12208
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