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Theorem elabgft1 13038
Description: One implication of elabgf 2826, 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 117 . . . . . 6 ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥𝜑} → 𝜑))
2 imim2 55 . . . . . 6 ((𝜑𝜓) → ((𝐴 ∈ {𝑥𝜑} → 𝜑) → (𝐴 ∈ {𝑥𝜑} → 𝜓)))
31, 2syl5 32 . . . . 5 ((𝜑𝜓) → ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥𝜑} → 𝜓)))
43imim2i 12 . . . 4 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥𝜑} → 𝜓))))
54alimi 1431 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥𝜑} → 𝜓))))
6 elabgf1.nf1 . . . 4 𝑥𝐴
7 nfab1 2283 . . . . . 6 𝑥{𝑥𝜑}
86, 7nfel 2290 . . . . 5 𝑥 𝐴 ∈ {𝑥𝜑}
9 elabgf1.nf2 . . . . 5 𝑥𝜓
108, 9nfim 1551 . . . 4 𝑥(𝐴 ∈ {𝑥𝜑} → 𝜓)
11 elabgf0 13037 . . . 4 (𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))
126, 10, 11bj-vtoclgft 13035 . . 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 50 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ {𝑥𝜑} → 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1329   = wceq 1331  wnf 1436  wcel 1480  {cab 2125  wnfc 2268
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by:  elabgf1  13039
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