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Theorem intabs 5209
 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 3940 . . . . . 6 (𝑥 = {𝑦𝜓} → (𝑥𝐴 {𝑦𝜓} ⊆ 𝐴))
2 intabs.2 . . . . . 6 (𝑥 = {𝑦𝜓} → (𝜑𝜒))
31, 2anbi12d 633 . . . . 5 (𝑥 = {𝑦𝜓} → ((𝑥𝐴𝜑) ↔ ( {𝑦𝜓} ⊆ 𝐴𝜒)))
4 intabs.3 . . . . 5 ( {𝑦𝜓} ⊆ 𝐴𝜒)
53, 4intmin3 4866 . . . 4 ( {𝑦𝜓} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
6 intnex 5205 . . . . 5 {𝑦𝜓} ∈ V ↔ {𝑦𝜓} = V)
7 ssv 3939 . . . . . 6 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ V
8 sseq2 3941 . . . . . 6 ( {𝑦𝜓} = V → ( {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓} ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ V))
97, 8mpbiri 261 . . . . 5 ( {𝑦𝜓} = V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
106, 9sylbi 220 . . . 4 {𝑦𝜓} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
115, 10pm2.61i 185 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓}
12 intabs.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
1312cbvabv 2866 . . . 4 {𝑥𝜑} = {𝑦𝜓}
1413inteqi 4842 . . 3 {𝑥𝜑} = {𝑦𝜓}
1511, 14sseqtrri 3952 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑}
16 simpr 488 . . . 4 ((𝑥𝐴𝜑) → 𝜑)
1716ss2abi 3994 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑}
18 intss 4859 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑} → {𝑥𝜑} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
1917, 18ax-mp 5 . 2 {𝑥𝜑} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)}
2015, 19eqssi 3931 1 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥𝜑}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  Vcvv 3441   ⊆ wss 3881  ∩ cint 4838 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-nul 4244  df-int 4839 This theorem is referenced by:  dfnn3  11641
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