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

Theorem sbcabel 3709
Description: Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
Hypothesis
Ref Expression
sbcabel.1 𝑥𝐵
Assertion
Ref Expression
sbcabel (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵 ↔ {𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcabel
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elex 3405 . 2 (𝐴𝑉𝐴 ∈ V)
2 sbcex2 3681 . . . 4 ([𝐴 / 𝑥]𝑤(𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ ∃𝑤[𝐴 / 𝑥](𝑤 = {𝑦𝜑} ∧ 𝑤𝐵))
3 sbcan 3673 . . . . . 6 ([𝐴 / 𝑥](𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ ([𝐴 / 𝑥]𝑤 = {𝑦𝜑} ∧ [𝐴 / 𝑥]𝑤𝐵))
4 sbcal 3680 . . . . . . . . 9 ([𝐴 / 𝑥]𝑦(𝑦𝑤𝜑) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝑤𝜑))
5 sbcbig 3675 . . . . . . . . . . 11 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑦𝑤𝜑) ↔ ([𝐴 / 𝑥]𝑦𝑤[𝐴 / 𝑥]𝜑)))
6 sbcg 3696 . . . . . . . . . . . 12 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝑤𝑦𝑤))
76bibi1d 334 . . . . . . . . . . 11 (𝐴 ∈ V → (([𝐴 / 𝑥]𝑦𝑤[𝐴 / 𝑥]𝜑) ↔ (𝑦𝑤[𝐴 / 𝑥]𝜑)))
85, 7bitrd 270 . . . . . . . . . 10 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑦𝑤𝜑) ↔ (𝑦𝑤[𝐴 / 𝑥]𝜑)))
98albidv 2014 . . . . . . . . 9 (𝐴 ∈ V → (∀𝑦[𝐴 / 𝑥](𝑦𝑤𝜑) ↔ ∀𝑦(𝑦𝑤[𝐴 / 𝑥]𝜑)))
104, 9syl5bb 274 . . . . . . . 8 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦(𝑦𝑤𝜑) ↔ ∀𝑦(𝑦𝑤[𝐴 / 𝑥]𝜑)))
11 abeq2 2915 . . . . . . . . 9 (𝑤 = {𝑦𝜑} ↔ ∀𝑦(𝑦𝑤𝜑))
1211sbcbii 3686 . . . . . . . 8 ([𝐴 / 𝑥]𝑤 = {𝑦𝜑} ↔ [𝐴 / 𝑥]𝑦(𝑦𝑤𝜑))
13 abeq2 2915 . . . . . . . 8 (𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ↔ ∀𝑦(𝑦𝑤[𝐴 / 𝑥]𝜑))
1410, 12, 133bitr4g 305 . . . . . . 7 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤 = {𝑦𝜑} ↔ 𝑤 = {𝑦[𝐴 / 𝑥]𝜑}))
15 sbcabel.1 . . . . . . . . 9 𝑥𝐵
1615nfcri 2941 . . . . . . . 8 𝑥 𝑤𝐵
1716sbcgf 3694 . . . . . . 7 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤𝐵𝑤𝐵))
1814, 17anbi12d 618 . . . . . 6 (𝐴 ∈ V → (([𝐴 / 𝑥]𝑤 = {𝑦𝜑} ∧ [𝐴 / 𝑥]𝑤𝐵) ↔ (𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵)))
193, 18syl5bb 274 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ (𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵)))
2019exbidv 2015 . . . 4 (𝐴 ∈ V → (∃𝑤[𝐴 / 𝑥](𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ ∃𝑤(𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵)))
212, 20syl5bb 274 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤(𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ ∃𝑤(𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵)))
22 df-clel 2801 . . . 4 ({𝑦𝜑} ∈ 𝐵 ↔ ∃𝑤(𝑤 = {𝑦𝜑} ∧ 𝑤𝐵))
2322sbcbii 3686 . . 3 ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵[𝐴 / 𝑥]𝑤(𝑤 = {𝑦𝜑} ∧ 𝑤𝐵))
24 df-clel 2801 . . 3 ({𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵 ↔ ∃𝑤(𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵))
2521, 23, 243bitr4g 305 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵 ↔ {𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵))
261, 25syl 17 1 (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵 ↔ {𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wex 1859  wcel 2158  {cab 2791  wnfc 2934  Vcvv 3390  [wsbc 3630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-v 3392  df-sbc 3631
This theorem is referenced by:  csbexg  4984
  Copyright terms: Public domain W3C validator