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

Theorem sb2 2350
Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2427) or a non-freeness hypothesis (sb6f 2383). (Contributed by NM, 13-May-1993.)
Assertion
Ref Expression
sb2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Proof of Theorem sb2
StepHypRef Expression
1 sp 2051 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 equs4 2288 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 df-sb 1879 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 697 1 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1479  wex 1702  [wsb 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-sb 1879
This theorem is referenced by:  stdpc4  2351  sb3  2353  sb4b  2356  hbsb2  2357  hbsb2a  2359  hbsb2e  2361  equsb1  2366  equsb2  2367  dfsb2  2371  sbequi  2373  sb6f  2383  sbi1  2390  sb6  2427  iota4  5857  wl-lem-moexsb  33321  sbeqal1  38418
  Copyright terms: Public domain W3C validator