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Theorem elab3gf 2686
 Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2679. (Contributed by NM, 6-Sep-2011.)
Hypotheses
Ref Expression
elab3gf.1 xA
elab3gf.2 xψ
elab3gf.3 (x = A → (φψ))
Assertion
Ref Expression
elab3gf ((ψA B) → (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 2679 . . 3 (A {xφ} → (A {xφ} ↔ ψ))
54ibi 165 . 2 (A {xφ} → ψ)
61, 2, 3elabgf 2679 . . . 4 (A B → (A {xφ} ↔ ψ))
76imim2i 12 . . 3 ((ψA B) → (ψ → (A {xφ} ↔ ψ)))
8 bi2 121 . . 3 ((A {xφ} ↔ ψ) → (ψA {xφ}))
97, 8syli 33 . 2 ((ψA B) → (ψA {xφ}))
105, 9impbid2 131 1 ((ψA B) → (A {xφ} ↔ ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  Ⅎwnf 1346   ∈ wcel 1390  {cab 2023  Ⅎwnfc 2162 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553 This theorem is referenced by:  elab3g  2687
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