ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcbi2 GIF version

Theorem sbcbi2 2836
Description: Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
Assertion
Ref Expression
sbcbi2 (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Proof of Theorem sbcbi2
StepHypRef Expression
1 abbi 2167 . . 3 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
2 eleq2 2117 . . 3 ({𝑥𝜑} = {𝑥𝜓} → (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜓}))
31, 2sylbi 118 . 2 (∀𝑥(𝜑𝜓) → (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜓}))
4 df-sbc 2788 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
5 df-sbc 2788 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
63, 4, 53bitr4g 216 1 (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259  wcel 1409  {cab 2042  [wsbc 2787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-sbc 2788
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator