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Theorem elabgf2 9188
Description: One implication of elabgf 2679. (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 2177 . . . 4 x{xφ}
41, 3nfel 2183 . . 3 x A {xφ}
52, 4nfim 1461 . 2 x(ψA {xφ})
6 elabgf0 9185 . 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 9184 1 (A B → (ψA {xφ}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wnf 1346   wcel 1390  {cab 2023  wnfc 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-bnd 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:  elabf2  9190  elabg2  9193  bj-intabssel1  9198
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