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Theorem sbciegft 3448
Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3449.) (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 3442 . . 3 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
2 biimp 205 . . . . . . . 8 ((𝜑𝜓) → (𝜑𝜓))
32imim2i 16 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
43impd 447 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → ((𝑥 = 𝐴𝜑) → 𝜓))
54alimi 1736 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥((𝑥 = 𝐴𝜑) → 𝜓))
6 19.23t 2077 . . . . . 6 (Ⅎ𝑥𝜓 → (∀𝑥((𝑥 = 𝐴𝜑) → 𝜓) ↔ (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓)))
76biimpa 501 . . . . 5 ((Ⅎ𝑥𝜓 ∧ ∀𝑥((𝑥 = 𝐴𝜑) → 𝜓)) → (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓))
85, 7sylan2 491 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓))
983adant1 1077 . . 3 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓))
101, 9syl5bi 232 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
11 biimpr 210 . . . . . . . 8 ((𝜑𝜓) → (𝜓𝜑))
1211imim2i 16 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜓𝜑)))
1312com23 86 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜓 → (𝑥 = 𝐴𝜑)))
1413alimi 1736 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)))
15 19.21t 2071 . . . . . 6 (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))))
1615biimpa 501 . . . . 5 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜓 → (𝑥 = 𝐴𝜑))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
1714, 16sylan2 491 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
18173adant1 1077 . . 3 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
19 sbc6g 3443 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
20193ad2ant1 1080 . . 3 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
2118, 20sylibrd 249 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓[𝐴 / 𝑥]𝜑))
2210, 21impbid 202 1 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wnf 1705  wcel 1987  [wsbc 3417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3188  df-sbc 3418
This theorem is referenced by:  sbciegf  3449  sbciedf  3453
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