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Theorem elabgft1 9917
Description: One implication of elabgf 2685, 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 1344 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥𝜑} → 𝜓))))
6 elabgf1.nf1 . . . 4 𝑥𝐴
7 nfab1 2180 . . . . . 6 𝑥{𝑥𝜑}
86, 7nfel 2186 . . . . 5 𝑥 𝐴 ∈ {𝑥𝜑}
9 elabgf1.nf2 . . . . 5 𝑥𝜓
108, 9nfim 1464 . . . 4 𝑥(𝐴 ∈ {𝑥𝜑} → 𝜓)
11 elabgf0 9916 . . . 4 (𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))
126, 10, 11bj-vtoclgft 9914 . . 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 1241   = 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:  elabgf1  9918
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