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Theorem sbcrext 3544
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
Assertion
Ref Expression
sbcrext (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 sbcrext
StepHypRef Expression
1 sbcex 3478 . . 3 ([𝐴 / 𝑥]𝑦𝐵 𝜑𝐴 ∈ V)
21a1i 11 . 2 (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑𝐴 ∈ V))
3 nfnfc1 2796 . . 3 𝑦𝑦𝐴
4 id 22 . . . 4 (𝑦𝐴𝑦𝐴)
5 nfcvd 2794 . . . 4 (𝑦𝐴𝑦V)
64, 5nfeld 2802 . . 3 (𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V)
7 sbcex 3478 . . . 4 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
872a1i 12 . . 3 (𝑦𝐴 → (𝑦𝐵 → ([𝐴 / 𝑥]𝜑𝐴 ∈ V)))
93, 6, 8rexlimd2 3054 . 2 (𝑦𝐴 → (∃𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V))
10 sbcng 3509 . . . . . 6 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑))
1110adantl 481 . . . . 5 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑))
12 sbcralt 3543 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑))
1312ancoms 468 . . . . . . 7 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑))
143, 6nfan1 2106 . . . . . . . 8 𝑦(𝑦𝐴𝐴 ∈ V)
15 sbcng 3509 . . . . . . . . 9 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
1615adantl 481 . . . . . . . 8 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
1714, 16ralbid 3012 . . . . . . 7 ((𝑦𝐴𝐴 ∈ V) → (∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
1813, 17bitrd 268 . . . . . 6 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
1918notbid 307 . . . . 5 ((𝑦𝐴𝐴 ∈ V) → (¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
2011, 19bitrd 268 . . . 4 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
21 dfrex2 3025 . . . . 5 (∃𝑦𝐵 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ 𝜑)
2221sbcbii 3524 . . . 4 ([𝐴 / 𝑥]𝑦𝐵 𝜑[𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑)
23 dfrex2 3025 . . . 4 (∃𝑦𝐵 [𝐴 / 𝑥]𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑)
2420, 22, 233bitr4g 303 . . 3 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2524ex 449 . 2 (𝑦𝐴 → (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)))
262, 9, 25pm5.21ndd 368 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 383  wcel 2030  wnfc 2780  wral 2941  wrex 2942  Vcvv 3231  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-sbc 3469
This theorem is referenced by:  sbcrex  3547
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