Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  spimv1 Structured version   Visualization version   GIF version

Theorem spimv1 2238
 Description: Version of spim 2351 with a disjoint variable condition, which does not require ax-13 2333. See spimvw 2045 for a version with two disjoint variable conditions, requiring fewer axioms, and spimv 2354 for another variant. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
spimv1.nf 𝑥𝜓
spimv1.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimv1 (∀𝑥𝜑𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem spimv1
StepHypRef Expression
1 spimv1.nf . 2 𝑥𝜓
2 ax6ev 2023 . . 3 𝑥 𝑥 = 𝑦
3 spimv1.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3eximii 1880 . 2 𝑥(𝜑𝜓)
51, 419.36i 2216 1 (∀𝑥𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1599  Ⅎwnf 1827 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-12 2162 This theorem depends on definitions:  df-bi 199  df-ex 1824  df-nf 1828 This theorem is referenced by:  cbv3v2  2239  cbv3v  2309  bj-chvarv  33327
 Copyright terms: Public domain W3C validator