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Theorem sbciegft 2869
Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2870.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbciegft ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem sbciegft
StepHypRef Expression
1 sbc5 2863 . . 3 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
2 bi1 116 . . . . . . . 8 ((𝜑𝜓) → (𝜑𝜓))
32imim2i 12 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
43impd 251 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → ((𝑥 = 𝐴𝜑) → 𝜓))
54alimi 1389 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥((𝑥 = 𝐴𝜑) → 𝜓))
6 19.23t 1612 . . . . . 6 (Ⅎ𝑥𝜓 → (∀𝑥((𝑥 = 𝐴𝜑) → 𝜓) ↔ (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓)))
76biimpa 290 . . . . 5 ((Ⅎ𝑥𝜓 ∧ ∀𝑥((𝑥 = 𝐴𝜑) → 𝜓)) → (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓))
85, 7sylan2 280 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓))
983adant1 961 . . 3 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓))
101, 9syl5bi 150 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
11 bi2 128 . . . . . . . 8 ((𝜑𝜓) → (𝜓𝜑))
1211imim2i 12 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜓𝜑)))
1312com23 77 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜓 → (𝑥 = 𝐴𝜑)))
1413alimi 1389 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)))
15 19.21t 1519 . . . . . 6 (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))))
1615biimpa 290 . . . . 5 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜓 → (𝑥 = 𝐴𝜑))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
1714, 16sylan2 280 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
18173adant1 961 . . 3 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
19 sbc6g 2864 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
20193ad2ant1 964 . . 3 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
2118, 20sylibrd 167 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓[𝐴 / 𝑥]𝜑))
2210, 21impbid 127 1 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 924  wal 1287   = wceq 1289  wnf 1394  wex 1426  wcel 1438  [wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841
This theorem is referenced by:  sbciegf  2870  sbciedf  2874
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