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Theorem elabgft1 10276
Description: One implication of elabgf 2707, 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 115 . . . . . 6 ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥𝜑} → 𝜑))
2 imim2 53 . . . . . 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 1360 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝐴 ∈ {𝑥𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥𝜑} → 𝜓))))
6 elabgf1.nf1 . . . 4 𝑥𝐴
7 nfab1 2196 . . . . . 6 𝑥{𝑥𝜑}
86, 7nfel 2202 . . . . 5 𝑥 𝐴 ∈ {𝑥𝜑}
9 elabgf1.nf2 . . . . 5 𝑥𝜓
108, 9nfim 1480 . . . 4 𝑥(𝐴 ∈ {𝑥𝜑} → 𝜓)
11 elabgf0 10275 . . . 4 (𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))
126, 10, 11bj-vtoclgft 10273 . . 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 48 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ {𝑥𝜑} → 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = 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:  elabgf1  10277
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