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Theorem intmin4 4476
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 4464 . . . 4 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
2 simpr 477 . . . . . . . 8 ((𝐴𝑥𝜑) → 𝜑)
3 ancr 571 . . . . . . . 8 ((𝜑𝐴𝑥) → (𝜑 → (𝐴𝑥𝜑)))
42, 3impbid2 216 . . . . . . 7 ((𝜑𝐴𝑥) → ((𝐴𝑥𝜑) ↔ 𝜑))
54imbi1d 331 . . . . . 6 ((𝜑𝐴𝑥) → (((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)))
65alimi 1736 . . . . 5 (∀𝑥(𝜑𝐴𝑥) → ∀𝑥(((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)))
7 albi 1743 . . . . 5 (∀𝑥(((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)) → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
86, 7syl 17 . . . 4 (∀𝑥(𝜑𝐴𝑥) → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
91, 8sylbi 207 . . 3 (𝐴 {𝑥𝜑} → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
10 vex 3192 . . . 4 𝑦 ∈ V
1110elintab 4457 . . 3 (𝑦 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ ∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥))
1210elintab 4457 . . 3 (𝑦 {𝑥𝜑} ↔ ∀𝑥(𝜑𝑦𝑥))
139, 11, 123bitr4g 303 . 2 (𝐴 {𝑥𝜑} → (𝑦 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ 𝑦 {𝑥𝜑}))
1413eqrdv 2619 1 (𝐴 {𝑥𝜑} → {𝑥 ∣ (𝐴𝑥𝜑)} = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wss 3559   cint 4445
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
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-ral 2912  df-v 3191  df-in 3566  df-ss 3573  df-int 4446
This theorem is referenced by: (None)
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