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Theorem intmin4 3690
Description: Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
Assertion
Ref Expression
intmin4 (𝐴 {𝑥𝜑} → {𝑥 ∣ (𝐴𝑥𝜑)} = {𝑥𝜑})
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem intmin4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssintab 3679 . . . 4 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
2 simpr 108 . . . . . . . 8 ((𝐴𝑥𝜑) → 𝜑)
3 ancr 314 . . . . . . . 8 ((𝜑𝐴𝑥) → (𝜑 → (𝐴𝑥𝜑)))
42, 3impbid2 141 . . . . . . 7 ((𝜑𝐴𝑥) → ((𝐴𝑥𝜑) ↔ 𝜑))
54imbi1d 229 . . . . . 6 ((𝜑𝐴𝑥) → (((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)))
65alimi 1385 . . . . 5 (∀𝑥(𝜑𝐴𝑥) → ∀𝑥(((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)))
7 albi 1398 . . . . 5 (∀𝑥(((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)) → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
86, 7syl 14 . . . 4 (∀𝑥(𝜑𝐴𝑥) → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
91, 8sylbi 119 . . 3 (𝐴 {𝑥𝜑} → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
10 vex 2615 . . . 4 𝑦 ∈ V
1110elintab 3673 . . 3 (𝑦 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ ∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥))
1210elintab 3673 . . 3 (𝑦 {𝑥𝜑} ↔ ∀𝑥(𝜑𝑦𝑥))
139, 11, 123bitr4g 221 . 2 (𝐴 {𝑥𝜑} → (𝑦 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ 𝑦 {𝑥𝜑}))
1413eqrdv 2081 1 (𝐴 {𝑥𝜑} → {𝑥 ∣ (𝐴𝑥𝜑)} = {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wcel 1434  {cab 2069  wss 2984   cint 3662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-in 2990  df-ss 2997  df-int 3663
This theorem is referenced by: (None)
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