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Theorem sbciegft 2815
Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2816.) (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 2809 . . 3 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
2 bi1 115 . . . . . . . 8 ((𝜑𝜓) → (𝜑𝜓))
32imim2i 12 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
43impd 246 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → ((𝑥 = 𝐴𝜑) → 𝜓))
54alimi 1360 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥((𝑥 = 𝐴𝜑) → 𝜓))
6 19.23t 1583 . . . . . 6 (Ⅎ𝑥𝜓 → (∀𝑥((𝑥 = 𝐴𝜑) → 𝜓) ↔ (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓)))
76biimpa 284 . . . . 5 ((Ⅎ𝑥𝜓 ∧ ∀𝑥((𝑥 = 𝐴𝜑) → 𝜓)) → (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓))
85, 7sylan2 274 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓))
983adant1 933 . . 3 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓))
101, 9syl5bi 145 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
11 bi2 125 . . . . . . . 8 ((𝜑𝜓) → (𝜓𝜑))
1211imim2i 12 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜓𝜑)))
1312com23 76 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜓 → (𝑥 = 𝐴𝜑)))
1413alimi 1360 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)))
15 19.21t 1490 . . . . . 6 (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))))
1615biimpa 284 . . . . 5 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜓 → (𝑥 = 𝐴𝜑))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
1714, 16sylan2 274 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
18173adant1 933 . . 3 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
19 sbc6g 2810 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
20193ad2ant1 936 . . 3 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
2118, 20sylibrd 162 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓[𝐴 / 𝑥]𝜑))
2210, 21impbid 124 1 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896  wal 1257   = wceq 1259  wnf 1365  wex 1397  wcel 1409  [wsbc 2786
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-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2787
This theorem is referenced by:  sbciegf  2816  sbciedf  2820
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