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Theorem elabgf2 12789
Description: One implication of elabgf 2798. (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 2258 . . . 4 𝑥{𝑥𝜑}
41, 3nfel 2265 . . 3 𝑥 𝐴 ∈ {𝑥𝜑}
52, 4nfim 1534 . 2 𝑥(𝜓𝐴 ∈ {𝑥𝜑})
6 elabgf0 12786 . 2 (𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))
7 bicom1 130 . . 3 ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝜑𝐴 ∈ {𝑥𝜑}))
8 elabgf2.1 . . . 4 (𝑥 = 𝐴 → (𝜓𝜑))
9 bi1 117 . . . 4 ((𝜑𝐴 ∈ {𝑥𝜑}) → (𝜑𝐴 ∈ {𝑥𝜑}))
108, 9syl9 72 . . 3 (𝑥 = 𝐴 → ((𝜑𝐴 ∈ {𝑥𝜑}) → (𝜓𝐴 ∈ {𝑥𝜑})))
117, 10syl5 32 . 2 (𝑥 = 𝐴 → ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝜓𝐴 ∈ {𝑥𝜑})))
121, 5, 6, 11bj-vtoclgf 12785 1 (𝐴𝐵 → (𝜓𝐴 ∈ {𝑥𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wnf 1419  wcel 1463  {cab 2101  wnfc 2243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660
This theorem is referenced by:  elabf2  12791  elabg2  12794  bj-intabssel1  12799
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