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Theorem elabgf2 7026
 Description: One implication of elabgf 2662. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
elabgf2.nf1 xA
elabgf2.nf2 xψ
elabgf2.1 (x = A → (ψφ))
Assertion
Ref Expression
elabgf2 (A B → (ψA {xφ}))

Proof of Theorem elabgf2
StepHypRef Expression
1 elabgf2.nf1 . 2 xA
2 elabgf2.nf2 . . 3 xψ
3 nfab1 2162 . . . 4 x{xφ}
41, 3nfel 2168 . . 3 x A {xφ}
52, 4nfim 1446 . 2 x(ψA {xφ})
6 elabgf0 7023 . 2 (x = A → (A {xφ} ↔ φ))
7 bicom1 122 . . 3 ((A {xφ} ↔ φ) → (φA {xφ}))
8 elabgf2.1 . . . 4 (x = A → (ψφ))
9 bi1 111 . . . 4 ((φA {xφ}) → (φA {xφ}))
108, 9syl9 66 . . 3 (x = A → ((φA {xφ}) → (ψA {xφ})))
117, 10syl5 28 . 2 (x = A → ((A {xφ} ↔ φ) → (ψA {xφ})))
121, 5, 6, 11bj-vtoclgf 7022 1 (A B → (ψA {xφ}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228  Ⅎwnf 1329   ∈ wcel 1374  {cab 2008  Ⅎwnfc 2147 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-v 2537 This theorem is referenced by:  elabf2  7028  elabg2  7031  bj-intabssel1  7036
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