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Theorem sbcrextOLD 3498
Description: Obsolete proof of sbcrext 3497 as of 7-Jul-2021. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcrextOLD (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
Distinct variable groups:   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem sbcrextOLD
StepHypRef Expression
1 sbcng 3462 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑))
21adantr 481 . . . 4 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑))
3 sbcralt 3496 . . . . . 6 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑))
4 nfnfc1 2764 . . . . . . . . 9 𝑦𝑦𝐴
5 id 22 . . . . . . . . . 10 (𝑦𝐴𝑦𝐴)
6 nfcvd 2762 . . . . . . . . . 10 (𝑦𝐴𝑦V)
75, 6nfeld 2769 . . . . . . . . 9 (𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V)
84, 7nfan1 2066 . . . . . . . 8 𝑦(𝑦𝐴𝐴 ∈ V)
9 sbcng 3462 . . . . . . . . 9 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
109adantl 482 . . . . . . . 8 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
118, 10ralbid 2978 . . . . . . 7 ((𝑦𝐴𝐴 ∈ V) → (∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
1211ancoms 469 . . . . . 6 ((𝐴 ∈ V ∧ 𝑦𝐴) → (∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
133, 12bitrd 268 . . . . 5 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
1413notbid 308 . . . 4 ((𝐴 ∈ V ∧ 𝑦𝐴) → (¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
152, 14bitrd 268 . . 3 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
16 dfrex2 2991 . . . 4 (∃𝑦𝐵 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ 𝜑)
1716sbcbii 3477 . . 3 ([𝐴 / 𝑥]𝑦𝐵 𝜑[𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑)
18 dfrex2 2991 . . 3 (∃𝑦𝐵 [𝐴 / 𝑥]𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑)
1915, 17, 183bitr4g 303 . 2 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
20 sbcex 3431 . . . . 5 ([𝐴 / 𝑥]𝑦𝐵 𝜑𝐴 ∈ V)
2120con3i 150 . . . 4 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵 𝜑)
2221adantr 481 . . 3 ((¬ 𝐴 ∈ V ∧ 𝑦𝐴) → ¬ [𝐴 / 𝑥]𝑦𝐵 𝜑)
23 sbcex 3431 . . . . . . 7 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
24232a1i 12 . . . . . 6 (𝑦𝐴 → (𝑦𝐵 → ([𝐴 / 𝑥]𝜑𝐴 ∈ V)))
254, 7, 24rexlimd2 3019 . . . . 5 (𝑦𝐴 → (∃𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V))
2625con3rr3 151 . . . 4 𝐴 ∈ V → (𝑦𝐴 → ¬ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2726imp 445 . . 3 ((¬ 𝐴 ∈ V ∧ 𝑦𝐴) → ¬ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)
2822, 272falsed 366 . 2 ((¬ 𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2919, 28pm2.61ian 830 1 (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wcel 1987  wnfc 2748  wral 2907  wrex 2908  Vcvv 3189  [wsbc 3421
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-11 2031  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-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-sbc 3422
This theorem is referenced by: (None)
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