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Theorem elabgf2 9919
Description: One implication of elabgf 2685. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
elabgf2.nf1 𝑥𝐴
elabgf2.nf2 𝑥𝜓
elabgf2.1 (𝑥 = 𝐴 → (𝜓𝜑))
Assertion
Ref Expression
elabgf2 (𝐴𝐵 → (𝜓𝐴 ∈ {𝑥𝜑}))

Proof of Theorem elabgf2
StepHypRef Expression
1 elabgf2.nf1 . 2 𝑥𝐴
2 elabgf2.nf2 . . 3 𝑥𝜓
3 nfab1 2180 . . . 4 𝑥{𝑥𝜑}
41, 3nfel 2186 . . 3 𝑥 𝐴 ∈ {𝑥𝜑}
52, 4nfim 1464 . 2 𝑥(𝜓𝐴 ∈ {𝑥𝜑})
6 elabgf0 9916 . 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 9915 1 (𝐴𝐵 → (𝜓𝐴 ∈ {𝑥𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wnf 1349  wcel 1393  {cab 2026  wnfc 2165
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559
This theorem is referenced by:  elabf2  9921  elabg2  9924  bj-intabssel1  9929
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