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

Theorem sbhypf 3558
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3915. (Contributed by Raph Levien, 10-Apr-2004.)
Hypotheses
Ref Expression
sbhypf.1 𝑥𝜓
sbhypf.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
sbhypf (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem sbhypf
StepHypRef Expression
1 eqeq1 2830 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
21equsexvw 2004 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
3 nfs1v 2267 . . . 4 𝑥[𝑦 / 𝑥]𝜑
4 sbhypf.1 . . . 4 𝑥𝜓
53, 4nfbi 1897 . . 3 𝑥([𝑦 / 𝑥]𝜑𝜓)
6 sbequ12 2246 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
76bicomd 224 . . . 4 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
8 sbhypf.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
97, 8sylan9bb 510 . . 3 ((𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
105, 9exlimi 2210 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
112, 10sylbir 236 1 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wnf 1777  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-9 2117  ax-10 2138  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-cleq 2819
This theorem is referenced by:  mob2  3710  reu2eqd  3731  cbvrabcsfw  3928  cbvopab1  5136  ralxpf  5716  cbviotaw  6319  cbvriotaw  7115  tfisi  7561  ac6sf  9900  nn0ind-raph  12071  ac6sf2  30288  nn0min  30453  ac6gf  34878  fdc1  34892
  Copyright terms: Public domain W3C validator