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Theorem intabs 5355
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 4021 . . . . . 6 (𝑥 = {𝑦𝜓} → (𝑥𝐴 {𝑦𝜓} ⊆ 𝐴))
2 intabs.2 . . . . . 6 (𝑥 = {𝑦𝜓} → (𝜑𝜒))
31, 2anbi12d 632 . . . . 5 (𝑥 = {𝑦𝜓} → ((𝑥𝐴𝜑) ↔ ( {𝑦𝜓} ⊆ 𝐴𝜒)))
4 intabs.3 . . . . 5 ( {𝑦𝜓} ⊆ 𝐴𝜒)
53, 4intmin3 4981 . . . 4 ( {𝑦𝜓} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
6 intnex 5351 . . . . 5 {𝑦𝜓} ∈ V ↔ {𝑦𝜓} = V)
7 ssv 4020 . . . . . 6 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ V
8 sseq2 4022 . . . . . 6 ( {𝑦𝜓} = V → ( {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓} ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ V))
97, 8mpbiri 258 . . . . 5 ( {𝑦𝜓} = V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
106, 9sylbi 217 . . . 4 {𝑦𝜓} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
115, 10pm2.61i 182 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓}
12 intabs.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
1312cbvabv 2810 . . . 4 {𝑥𝜑} = {𝑦𝜓}
1413inteqi 4955 . . 3 {𝑥𝜑} = {𝑦𝜓}
1511, 14sseqtrri 4033 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑}
16 simpr 484 . . . 4 ((𝑥𝐴𝜑) → 𝜑)
1716ss2abi 4077 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑}
18 intss 4974 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑} → {𝑥𝜑} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
1917, 18ax-mp 5 . 2 {𝑥𝜑} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)}
2015, 19eqssi 4012 1 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478  wss 3963   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-int 4952
This theorem is referenced by:  dfnn3  12278
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