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

Theorem sbiedwOLD 2326
 Description: Obsolete version of sbiedw 2325 as of 28-Jan-2024. (Contributed by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
sbiedw.1 𝑥𝜑
sbiedw.2 (𝜑 → Ⅎ𝑥𝜒)
sbiedw.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
sbiedwOLD (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem sbiedwOLD
StepHypRef Expression
1 sbiedw.1 . . . 4 𝑥𝜑
21sbrim 2306 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
3 sbiedw.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
41, 3nfim1 2191 . . . 4 𝑥(𝜑𝜒)
5 sbiedw.3 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
65com12 32 . . . . 5 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
76pm5.74d 275 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
84, 7sbiev 2323 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑𝜒))
92, 8bitr3i 279 . 2 ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑𝜒))
109pm5.74ri 274 1 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  Ⅎwnf 1777  [wsb 2062 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator