Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1468 Structured version   Visualization version   GIF version

Theorem bnj1468 31244
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1468.1 (𝜓 → ∀𝑥𝜓)
bnj1468.2 (𝑥 = 𝐴 → (𝜑𝜓))
bnj1468.3 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Assertion
Ref Expression
bnj1468 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑉   𝜑,𝑦   𝜓,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem bnj1468
StepHypRef Expression
1 sbcco 3663 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
2 ax-5 2001 . . 3 (𝜓 → ∀𝑦𝜓)
3 bnj1468.3 . . . . . . . 8 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
43nfcii 2946 . . . . . . 7 𝑥𝐴
54nfeq2 2971 . . . . . 6 𝑥 𝑦 = 𝐴
6 nfsbc1v 3660 . . . . . . 7 𝑥[𝑦 / 𝑥]𝜑
7 bnj1468.1 . . . . . . . 8 (𝜓 → ∀𝑥𝜓)
87nf5i 2191 . . . . . . 7 𝑥𝜓
96, 8nfbi 1995 . . . . . 6 𝑥([𝑦 / 𝑥]𝜑𝜓)
105, 9nfim 1987 . . . . 5 𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
1110nf5ri 2231 . . . 4 ((𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓)) → ∀𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓)))
12 ax6ev 2071 . . . . 5 𝑥 𝑥 = 𝑦
13 eqeq1 2817 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
14 bnj1468.2 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
1513, 14syl6bir 245 . . . . . 6 (𝑥 = 𝑦 → (𝑦 = 𝐴 → (𝜑𝜓)))
16 sbceq1a 3651 . . . . . . 7 (𝑥 = 𝑦 → (𝜑[𝑦 / 𝑥]𝜑))
1716bibi1d 334 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))
1815, 17sylibd 230 . . . . 5 (𝑥 = 𝑦 → (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓)))
1912, 18bnj101 31120 . . . 4 𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
2011, 19bnj1131 31186 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
212, 20bnj1464 31242 . 2 (𝐴𝑉 → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑𝜓))
221, 21syl5bbr 276 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1635   = wceq 1637  wcel 2157  [wsbc 3640
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 2791
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 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-v 3400  df-sbc 3641
This theorem is referenced by:  bnj1463  31451
  Copyright terms: Public domain W3C validator