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Theorem sbciegft 2853
 Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2854.) (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 2847 . . 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 1385 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥((𝑥 = 𝐴𝜑) → 𝜓))
6 19.23t 1608 . . . . . 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 957 . . 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 1385 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)))
15 19.21t 1515 . . . . . 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 957 . . 3 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
19 sbc6g 2848 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
20193ad2ant1 960 . . 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 920  ∀wal 1283   = wceq 1285  Ⅎwnf 1390  ∃wex 1422   ∈ wcel 1434  [wsbc 2824 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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825 This theorem is referenced by:  sbciegf  2854  sbciedf  2858
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