MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbie2t Structured version   Visualization version   GIF version

Theorem csbie2t 3949
Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3950). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbie2t.1 𝐴 ∈ V
csbie2t.2 𝐵 ∈ V
Assertion
Ref Expression
csbie2t (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem csbie2t
StepHypRef Expression
1 nfa1 2149 . 2 𝑥𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
2 nfcvd 2904 . 2 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝑥𝐷)
3 csbie2t.1 . . 3 𝐴 ∈ V
43a1i 11 . 2 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 ∈ V)
5 nfa2 2174 . . . 4 𝑦𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
6 nfv 1912 . . . 4 𝑦 𝑥 = 𝐴
75, 6nfan 1897 . . 3 𝑦(∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴)
8 nfcvd 2904 . . 3 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → 𝑦𝐷)
9 csbie2t.2 . . . 4 𝐵 ∈ V
109a1i 11 . . 3 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
11 2sp 2184 . . . 4 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷))
1211impl 455 . . 3 (((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
137, 8, 10, 12csbiedf 3939 . 2 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → 𝐵 / 𝑦𝐶 = 𝐷)
141, 2, 4, 13csbiedf 3939 1 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  Vcvv 3478  csb 3908
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-3an 1088  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-v 3480  df-sbc 3792  df-csb 3909
This theorem is referenced by:  csbie2  3950
  Copyright terms: Public domain W3C validator