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Theorem intmin4 4696
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 4684 . . . 4 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
2 simpr 478 . . . . . . . 8 ((𝐴𝑥𝜑) → 𝜑)
3 ancr 543 . . . . . . . 8 ((𝜑𝐴𝑥) → (𝜑 → (𝐴𝑥𝜑)))
42, 3impbid2 218 . . . . . . 7 ((𝜑𝐴𝑥) → ((𝐴𝑥𝜑) ↔ 𝜑))
54imbi1d 333 . . . . . 6 ((𝜑𝐴𝑥) → (((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)))
65alimi 1907 . . . . 5 (∀𝑥(𝜑𝐴𝑥) → ∀𝑥(((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)))
7 albi 1914 . . . . 5 (∀𝑥(((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)) → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
86, 7syl 17 . . . 4 (∀𝑥(𝜑𝐴𝑥) → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
91, 8sylbi 209 . . 3 (𝐴 {𝑥𝜑} → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
10 vex 3388 . . . 4 𝑦 ∈ V
1110elintab 4678 . . 3 (𝑦 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ ∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥))
1210elintab 4678 . . 3 (𝑦 {𝑥𝜑} ↔ ∀𝑥(𝜑𝑦𝑥))
139, 11, 123bitr4g 306 . 2 (𝐴 {𝑥𝜑} → (𝑦 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ 𝑦 {𝑥𝜑}))
1413eqrdv 2797 1 (𝐴 {𝑥𝜑} → {𝑥 ∣ (𝐴𝑥𝜑)} = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wal 1651   = wceq 1653  wcel 2157  {cab 2785  wss 3769   cint 4667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387  df-in 3776  df-ss 3783  df-int 4668
This theorem is referenced by: (None)
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