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Theorem elabgft1 13778
Description: One implication of elabgf 2872, 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 biimp 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 1448 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥𝜑} → 𝜓))))
6 elabgf1.nf1 . . . 4 𝑥𝐴
7 nfab1 2314 . . . . . 6 𝑥{𝑥𝜑}
86, 7nfel 2321 . . . . 5 𝑥 𝐴 ∈ {𝑥𝜑}
9 elabgf1.nf2 . . . . 5 𝑥𝜓
108, 9nfim 1565 . . . 4 𝑥(𝐴 ∈ {𝑥𝜑} → 𝜓)
11 elabgf0 13777 . . . 4 (𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))
126, 10, 11bj-vtoclgft 13775 . . 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 1346   = wceq 1348  wnf 1453  wcel 2141  {cab 2156  wnfc 2299
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732
This theorem is referenced by:  elabgf1  13779
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