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Theorem intmin4 4896
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 4884 . . . 4 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
2 simpr 487 . . . . . . . 8 ((𝐴𝑥𝜑) → 𝜑)
3 ancr 549 . . . . . . . 8 ((𝜑𝐴𝑥) → (𝜑 → (𝐴𝑥𝜑)))
42, 3impbid2 228 . . . . . . 7 ((𝜑𝐴𝑥) → ((𝐴𝑥𝜑) ↔ 𝜑))
54imbi1d 344 . . . . . 6 ((𝜑𝐴𝑥) → (((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)))
65alimi 1806 . . . . 5 (∀𝑥(𝜑𝐴𝑥) → ∀𝑥(((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)))
7 albi 1813 . . . . 5 (∀𝑥(((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)) → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
86, 7syl 17 . . . 4 (∀𝑥(𝜑𝐴𝑥) → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
91, 8sylbi 219 . . 3 (𝐴 {𝑥𝜑} → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
10 vex 3496 . . . 4 𝑦 ∈ V
1110elintab 4878 . . 3 (𝑦 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ ∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥))
1210elintab 4878 . . 3 (𝑦 {𝑥𝜑} ↔ ∀𝑥(𝜑𝑦𝑥))
139, 11, 123bitr4g 316 . 2 (𝐴 {𝑥𝜑} → (𝑦 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ 𝑦 {𝑥𝜑}))
1413eqrdv 2817 1 (𝐴 {𝑥𝜑} → {𝑥 ∣ (𝐴𝑥𝜑)} = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1529   = wceq 1531  wcel 2108  {cab 2797  wss 3934   cint 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-v 3495  df-in 3941  df-ss 3950  df-int 4868
This theorem is referenced by: (None)
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