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Theorem elabgft1 9186
Description: One implication of elabgf 2679, in closed form. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
elabgf1.nf1 xA
elabgf1.nf2 xψ
Assertion
Ref Expression
elabgft1 (x(x = A → (φψ)) → (A {xφ} → ψ))

Proof of Theorem elabgft1
StepHypRef Expression
1 bi1 111 . . . . . 6 ((A {xφ} ↔ φ) → (A {xφ} → φ))
2 imim2 49 . . . . . 6 ((φψ) → ((A {xφ} → φ) → (A {xφ} → ψ)))
31, 2syl5 28 . . . . 5 ((φψ) → ((A {xφ} ↔ φ) → (A {xφ} → ψ)))
43imim2i 12 . . . 4 ((x = A → (φψ)) → (x = A → ((A {xφ} ↔ φ) → (A {xφ} → ψ))))
54alimi 1341 . . 3 (x(x = A → (φψ)) → x(x = A → ((A {xφ} ↔ φ) → (A {xφ} → ψ))))
6 elabgf1.nf1 . . . 4 xA
7 nfab1 2177 . . . . . 6 x{xφ}
86, 7nfel 2183 . . . . 5 x A {xφ}
9 elabgf1.nf2 . . . . 5 xψ
108, 9nfim 1461 . . . 4 x(A {xφ} → ψ)
11 elabgf0 9185 . . . 4 (x = A → (A {xφ} ↔ φ))
126, 10, 11bj-vtoclgft 9183 . . 3 (x(x = A → ((A {xφ} ↔ φ) → (A {xφ} → ψ))) → (A {xφ} → (A {xφ} → ψ)))
135, 12syl 14 . 2 (x(x = A → (φψ)) → (A {xφ} → (A {xφ} → ψ)))
1413pm2.43d 44 1 (x(x = A → (φψ)) → (A {xφ} → ψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = 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:  elabgf1  9187
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