MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb2 Structured version   Visualization version   GIF version

Theorem sb2 2340
Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2417) or a non-freeness hypothesis (sb6f 2373). (Contributed by NM, 13-May-1993.)
Assertion
Ref Expression
sb2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Proof of Theorem sb2
StepHypRef Expression
1 sp 2041 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 equs4 2278 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 df-sb 1868 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 695 1 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1473  wex 1695  [wsb 1867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868
This theorem is referenced by:  stdpc4  2341  sb3  2343  sb4b  2346  hbsb2  2347  hbsb2a  2349  hbsb2e  2351  equsb1  2356  equsb2  2357  dfsb2  2361  sbequi  2363  sb6f  2373  sbi1  2380  sb6  2417  iota4  5786  wl-lem-moexsb  32529  sbeqal1  37620
  Copyright terms: Public domain W3C validator