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Theorem intmin4 3670
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 3659 . . . 4 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
2 simpr 107 . . . . . . . 8 ((𝐴𝑥𝜑) → 𝜑)
3 ancr 308 . . . . . . . 8 ((𝜑𝐴𝑥) → (𝜑 → (𝐴𝑥𝜑)))
42, 3impbid2 135 . . . . . . 7 ((𝜑𝐴𝑥) → ((𝐴𝑥𝜑) ↔ 𝜑))
54imbi1d 224 . . . . . 6 ((𝜑𝐴𝑥) → (((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)))
65alimi 1360 . . . . 5 (∀𝑥(𝜑𝐴𝑥) → ∀𝑥(((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)))
7 albi 1373 . . . . 5 (∀𝑥(((𝐴𝑥𝜑) → 𝑦𝑥) ↔ (𝜑𝑦𝑥)) → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
86, 7syl 14 . . . 4 (∀𝑥(𝜑𝐴𝑥) → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
91, 8sylbi 118 . . 3 (𝐴 {𝑥𝜑} → (∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥) ↔ ∀𝑥(𝜑𝑦𝑥)))
10 vex 2577 . . . 4 𝑦 ∈ V
1110elintab 3653 . . 3 (𝑦 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ ∀𝑥((𝐴𝑥𝜑) → 𝑦𝑥))
1210elintab 3653 . . 3 (𝑦 {𝑥𝜑} ↔ ∀𝑥(𝜑𝑦𝑥))
139, 11, 123bitr4g 216 . 2 (𝐴 {𝑥𝜑} → (𝑦 {𝑥 ∣ (𝐴𝑥𝜑)} ↔ 𝑦 {𝑥𝜑}))
1413eqrdv 2054 1 (𝐴 {𝑥𝜑} → {𝑥 ∣ (𝐴𝑥𝜑)} = {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wcel 1409  {cab 2042  wss 2944   cint 3642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2951  df-ss 2958  df-int 3643
This theorem is referenced by: (None)
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