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Theorem sbcrext 3882
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 3801 . . 3 ([𝐴 / 𝑥]𝑦𝐵 𝜑𝐴 ∈ V)
21a1i 11 . 2 (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑𝐴 ∈ V))
3 nfnfc1 2906 . . 3 𝑦𝑦𝐴
4 id 22 . . . 4 (𝑦𝐴𝑦𝐴)
5 nfcvd 2904 . . . 4 (𝑦𝐴𝑦V)
64, 5nfeld 2915 . . 3 (𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V)
7 sbcex 3801 . . . 4 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
872a1i 12 . . 3 (𝑦𝐴 → (𝑦𝐵 → ([𝐴 / 𝑥]𝜑𝐴 ∈ V)))
93, 6, 8rexlimd2 3263 . 2 (𝑦𝐴 → (∃𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V))
10 sbcng 3842 . . . . . 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 3881 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑))
1312ancoms 458 . . . . . . 7 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑))
143, 6nfan1 2198 . . . . . . . 8 𝑦(𝑦𝐴𝐴 ∈ V)
15 sbcng 3842 . . . . . . . . 9 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
1615adantl 481 . . . . . . . 8 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
1714, 16ralbid 3271 . . . . . . 7 ((𝑦𝐴𝐴 ∈ V) → (∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
1813, 17bitrd 279 . . . . . 6 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
1918notbid 318 . . . . 5 ((𝑦𝐴𝐴 ∈ V) → (¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
2011, 19bitrd 279 . . . 4 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
21 dfrex2 3071 . . . . 5 (∃𝑦𝐵 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ 𝜑)
2221sbcbii 3852 . . . 4 ([𝐴 / 𝑥]𝑦𝐵 𝜑[𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑)
23 dfrex2 3071 . . . 4 (∃𝑦𝐵 [𝐴 / 𝑥]𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑)
2420, 22, 233bitr4g 314 . . 3 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2524ex 412 . 2 (𝑦𝐴 → (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)))
262, 9, 25pm5.21ndd 379 1 (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wcel 2106  wnfc 2888  wral 3059  wrex 3068  Vcvv 3478  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-sbc 3792
This theorem is referenced by:  sbcrex  3884
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