ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbimi GIF version

Theorem sbimi 1647
Description: Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
Hypothesis
Ref Expression
sbimi.1 (𝜑𝜓)
Assertion
Ref Expression
sbimi ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)

Proof of Theorem sbimi
StepHypRef Expression
1 sbimi.1 . . . 4 (𝜑𝜓)
21imim2i 12 . . 3 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓))
31anim2i 324 . . . 4 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓))
43eximi 1491 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜓))
52, 4anim12i 321 . 2 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
6 df-sb 1646 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
7 df-sb 1646 . 2 ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
85, 6, 73imtr4i 190 1 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wex 1381  [wsb 1645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-sb 1646
This theorem is referenced by:  sbbii  1648  sb6f  1684  hbsb3  1689  sbidm  1731  sbco  1842  sbcocom  1844  elsb3  1852  elsb4  1853  sbalyz  1875  hbsb4t  1889  moimv  1966  oprcl  3573  peano1  4317  peano2  4318
  Copyright terms: Public domain W3C validator