MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb4 Structured version   Visualization version   GIF version

Theorem sb4 2239
Description: One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
sb4 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem sb4
StepHypRef Expression
1 sb1 1831 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 equs5 2234 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
31, 2syl5ib 229 1 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 378  wal 1466  wex 1692  [wsb 1828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 192  df-an 380  df-ex 1693  df-nf 1697  df-sb 1829
This theorem is referenced by:  sb4b  2241  hbsb2  2242  dfsb2  2256  sbequi  2258  sbi1  2275
  Copyright terms: Public domain W3C validator