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Theorem elabgf2 10306
Description: One implication of elabgf 2708. (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 2196 . . . 4 𝑥{𝑥𝜑}
41, 3nfel 2202 . . 3 𝑥 𝐴 ∈ {𝑥𝜑}
52, 4nfim 1480 . 2 𝑥(𝜓𝐴 ∈ {𝑥𝜑})
6 elabgf0 10303 . 2 (𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))
7 bicom1 126 . . 3 ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝜑𝐴 ∈ {𝑥𝜑}))
8 elabgf2.1 . . . 4 (𝑥 = 𝐴 → (𝜓𝜑))
9 bi1 115 . . . 4 ((𝜑𝐴 ∈ {𝑥𝜑}) → (𝜑𝐴 ∈ {𝑥𝜑}))
108, 9syl9 70 . . 3 (𝑥 = 𝐴 → ((𝜑𝐴 ∈ {𝑥𝜑}) → (𝜓𝐴 ∈ {𝑥𝜑})))
117, 10syl5 32 . 2 (𝑥 = 𝐴 → ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝜓𝐴 ∈ {𝑥𝜑})))
121, 5, 6, 11bj-vtoclgf 10302 1 (𝐴𝐵 → (𝜓𝐴 ∈ {𝑥𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wnf 1365  wcel 1409  {cab 2042  wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  elabf2  10308  elabg2  10311  bj-intabssel1  10316
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