MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcrext Structured version   Visualization version   GIF version

Theorem sbcrext 3718
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 3654 . . 3 ([𝐴 / 𝑥]𝑦𝐵 𝜑𝐴 ∈ V)
21a1i 11 . 2 (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑𝐴 ∈ V))
3 nfnfc1 2962 . . 3 𝑦𝑦𝐴
4 id 22 . . . 4 (𝑦𝐴𝑦𝐴)
5 nfcvd 2960 . . . 4 (𝑦𝐴𝑦V)
64, 5nfeld 2968 . . 3 (𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V)
7 sbcex 3654 . . . 4 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
872a1i 12 . . 3 (𝑦𝐴 → (𝑦𝐵 → ([𝐴 / 𝑥]𝜑𝐴 ∈ V)))
93, 6, 8rexlimd2 3224 . 2 (𝑦𝐴 → (∃𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V))
10 sbcng 3685 . . . . . 6 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑))
1110adantl 469 . . . . 5 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑))
12 sbcralt 3717 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑))
1312ancoms 448 . . . . . . 7 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑))
143, 6nfan1 2234 . . . . . . . 8 𝑦(𝑦𝐴𝐴 ∈ V)
15 sbcng 3685 . . . . . . . . 9 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
1615adantl 469 . . . . . . . 8 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
1714, 16ralbid 3182 . . . . . . 7 ((𝑦𝐴𝐴 ∈ V) → (∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
1813, 17bitrd 270 . . . . . 6 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
1918notbid 309 . . . . 5 ((𝑦𝐴𝐴 ∈ V) → (¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
2011, 19bitrd 270 . . . 4 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
21 dfrex2 3194 . . . . 5 (∃𝑦𝐵 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ 𝜑)
2221sbcbii 3700 . . . 4 ([𝐴 / 𝑥]𝑦𝐵 𝜑[𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑)
23 dfrex2 3194 . . . 4 (∃𝑦𝐵 [𝐴 / 𝑥]𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑)
2420, 22, 233bitr4g 305 . . 3 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2524ex 399 . 2 (𝑦𝐴 → (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)))
262, 9, 25pm5.21ndd 370 1 (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wcel 2157  wnfc 2946  wral 3107  wrex 3108  Vcvv 3402  [wsbc 3644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-v 3404  df-sbc 3645
This theorem is referenced by:  sbcrex  3720
  Copyright terms: Public domain W3C validator