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Theorem intmin4 3617
Description: Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
Assertion
Ref Expression
intmin4 (A {xφ} → {x ∣ (Ax φ)} = {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 3606 . . . 4 (A {xφ} ↔ x(φAx))
2 simpr 103 . . . . . . . 8 ((Ax φ) → φ)
3 ancr 304 . . . . . . . 8 ((φAx) → (φ → (Ax φ)))
42, 3impbid2 131 . . . . . . 7 ((φAx) → ((Ax φ) ↔ φ))
54imbi1d 220 . . . . . 6 ((φAx) → (((Ax φ) → y x) ↔ (φy x)))
65alimi 1324 . . . . 5 (x(φAx) → x(((Ax φ) → y x) ↔ (φy x)))
7 albi 1337 . . . . 5 (x(((Ax φ) → y x) ↔ (φy x)) → (x((Ax φ) → y x) ↔ x(φy x)))
86, 7syl 14 . . . 4 (x(φAx) → (x((Ax φ) → y x) ↔ x(φy x)))
91, 8sylbi 114 . . 3 (A {xφ} → (x((Ax φ) → y x) ↔ x(φy x)))
10 vex 2538 . . . 4 y V
1110elintab 3600 . . 3 (y {x ∣ (Ax φ)} ↔ x((Ax φ) → y x))
1210elintab 3600 . . 3 (y {xφ} ↔ x(φy x))
139, 11, 123bitr4g 212 . 2 (A {xφ} → (y {x ∣ (Ax φ)} ↔ y {xφ}))
1413eqrdv 2020 1 (A {xφ} → {x ∣ (Ax φ)} = {xφ})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226   = wceq 1228   wcel 1374  {cab 2008  wss 2894   cint 3589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-in 2901  df-ss 2908  df-int 3590
This theorem is referenced by: (None)
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