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

Theorem sbbib 2376
 Description: Reversal of substitution. (Contributed by AV, 6-Aug-2023.) (Proof shortened by Wolf Lammen, 4-Sep-2023.)
Hypotheses
Ref Expression
sbbib.y 𝑦𝜑
sbbib.x 𝑥𝜓
Assertion
Ref Expression
sbbib (∀𝑦([𝑦 / 𝑥]𝜑𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbbib
StepHypRef Expression
1 nfs1v 2156 . . 3 𝑥[𝑦 / 𝑥]𝜑
2 sbbib.x . . 3 𝑥𝜓
31, 2nfbi 1900 . 2 𝑥([𝑦 / 𝑥]𝜑𝜓)
4 sbbib.y . . 3 𝑦𝜑
5 nfs1v 2156 . . 3 𝑦[𝑥 / 𝑦]𝜓
64, 5nfbi 1900 . 2 𝑦(𝜑 ↔ [𝑥 / 𝑦]𝜓)
7 sbequ12r 2250 . . 3 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
8 sbequ12 2249 . . 3 (𝑦 = 𝑥 → (𝜓 ↔ [𝑥 / 𝑦]𝜓))
97, 8bibi12d 348 . 2 (𝑦 = 𝑥 → (([𝑦 / 𝑥]𝜑𝜓) ↔ (𝜑 ↔ [𝑥 / 𝑦]𝜓)))
103, 6, 9cbvalv1 2357 1 (∀𝑦([𝑦 / 𝑥]𝜑𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208  ∀wal 1531  Ⅎwnf 1780  [wsb 2065 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2157  ax-12 2173 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066 This theorem is referenced by:  sbbibvv  2377  dfich2  43607  dfich2ai  43608
 Copyright terms: Public domain W3C validator