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Theorem elabgf2 11563
Description: One implication of elabgf 2758. (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 2230 . . . 4 𝑥{𝑥𝜑}
41, 3nfel 2237 . . 3 𝑥 𝐴 ∈ {𝑥𝜑}
52, 4nfim 1509 . 2 𝑥(𝜓𝐴 ∈ {𝑥𝜑})
6 elabgf0 11560 . 2 (𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))
7 bicom1 129 . . 3 ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝜑𝐴 ∈ {𝑥𝜑}))
8 elabgf2.1 . . . 4 (𝑥 = 𝐴 → (𝜓𝜑))
9 bi1 116 . . . 4 ((𝜑𝐴 ∈ {𝑥𝜑}) → (𝜑𝐴 ∈ {𝑥𝜑}))
108, 9syl9 71 . . 3 (𝑥 = 𝐴 → ((𝜑𝐴 ∈ {𝑥𝜑}) → (𝜓𝐴 ∈ {𝑥𝜑})))
117, 10syl5 32 . 2 (𝑥 = 𝐴 → ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝜓𝐴 ∈ {𝑥𝜑})))
121, 5, 6, 11bj-vtoclgf 11559 1 (𝐴𝐵 → (𝜓𝐴 ∈ {𝑥𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wnf 1394  wcel 1438  {cab 2074  wnfc 2215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  elabf2  11565  elabg2  11568  bj-intabssel1  11573
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