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Theorem elab3gf 2714
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2707. (Contributed by NM, 6-Sep-2011.)
Hypotheses
Ref Expression
elab3gf.1 𝑥𝐴
elab3gf.2 𝑥𝜓
elab3gf.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elab3gf ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))

Proof of Theorem elab3gf
StepHypRef Expression
1 elab3gf.1 . . . 4 𝑥𝐴
2 elab3gf.2 . . . 4 𝑥𝜓
3 elab3gf.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3elabgf 2707 . . 3 (𝐴 ∈ {𝑥𝜑} → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
54ibi 169 . 2 (𝐴 ∈ {𝑥𝜑} → 𝜓)
61, 2, 3elabgf 2707 . . . 4 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
76imim2i 12 . . 3 ((𝜓𝐴𝐵) → (𝜓 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)))
8 bi2 125 . . 3 ((𝐴 ∈ {𝑥𝜑} ↔ 𝜓) → (𝜓𝐴 ∈ {𝑥𝜑}))
97, 8syli 37 . 2 ((𝜓𝐴𝐵) → (𝜓𝐴 ∈ {𝑥𝜑}))
105, 9impbid2 135 1 ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wnf 1365  wcel 1409  {cab 2042  wnfc 2181
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-v 2576
This theorem is referenced by:  elab3g  2715
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