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

Theorem spim 2252
 Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2252 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-2018.)
Hypotheses
Ref Expression
spim.1 𝑥𝜓
spim.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spim (∀𝑥𝜑𝜓)

Proof of Theorem spim
StepHypRef Expression
1 spim.1 . 2 𝑥𝜓
2 ax6e 2248 . . 3 𝑥 𝑥 = 𝑦
3 spim.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3eximii 1762 . 2 𝑥(𝜑𝜓)
51, 419.36i 2097 1 (∀𝑥𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1479  Ⅎwnf 1706 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-nf 1708 This theorem is referenced by:  spimvALT  2256  chvar  2260  cbv3  2263  setrec2fun  42204
 Copyright terms: Public domain W3C validator