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Theorem intabs 4785
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 3605 . . . . . 6 (𝑥 = {𝑦𝜓} → (𝑥𝐴 {𝑦𝜓} ⊆ 𝐴))
2 intabs.2 . . . . . 6 (𝑥 = {𝑦𝜓} → (𝜑𝜒))
31, 2anbi12d 746 . . . . 5 (𝑥 = {𝑦𝜓} → ((𝑥𝐴𝜑) ↔ ( {𝑦𝜓} ⊆ 𝐴𝜒)))
4 intabs.3 . . . . 5 ( {𝑦𝜓} ⊆ 𝐴𝜒)
53, 4intmin3 4470 . . . 4 ( {𝑦𝜓} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑦𝜓})
6 intnex 4781 . . . . 5 {𝑦𝜓} ∈ V ↔ {𝑦𝜓} = V)
7 ssv 3604 . . . . . 6 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ V
8 sseq2 3606 . . . . . 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 2744 . . . 4 {𝑥𝜑} = {𝑦𝜓}
1413inteqi 4444 . . 3 {𝑥𝜑} = {𝑦𝜓}
1511, 14sseqtr4i 3617 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑}
16 simpr 477 . . . 4 ((𝑥𝐴𝜑) → 𝜑)
1716ss2abi 3653 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑}
18 intss 4463 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝜑} → {𝑥𝜑} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
1917, 18ax-mp 5 . 2 {𝑥𝜑} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)}
2015, 19eqssi 3599 1 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3186  wss 3555   cint 4440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-int 4441
This theorem is referenced by:  dfnn3  10978
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