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Theorem elssabg 4041
Description: Membership in a class abstraction involving a subset. Unlike elabg 2801, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.)
Hypothesis
Ref Expression
elssabg.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elssabg (𝐵𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem elssabg
StepHypRef Expression
1 ssexg 4035 . . . 4 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
21expcom 115 . . 3 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
32adantrd 275 . 2 (𝐵𝑉 → ((𝐴𝐵𝜓) → 𝐴 ∈ V))
4 sseq1 3088 . . . 4 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
5 elssabg.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
64, 5anbi12d 462 . . 3 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
76elab3g 2806 . 2 (((𝐴𝐵𝜓) → 𝐴 ∈ V) → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))
83, 7syl 14 1 (𝐵𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  {cab 2101  Vcvv 2658  wss 3039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052
This theorem is referenced by: (None)
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