Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ichf Structured version   Visualization version   GIF version

Theorem ichf 43438
 Description: Setvar variables are interchangeable in a wff they are not free in. (Contributed by SN, 23-Nov-2023.)
Hypotheses
Ref Expression
ichf.1 𝑥𝜑
ichf.2 𝑦𝜑
Assertion
Ref Expression
ichf [𝑥𝑦]𝜑

Proof of Theorem ichf
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 ichf.2 . . . . . . . 8 𝑦𝜑
21sbf 2264 . . . . . . 7 ([𝑎 / 𝑦]𝜑𝜑)
32sbbii 2074 . . . . . 6 ([𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4 ichf.1 . . . . . . 7 𝑥𝜑
54sbf 2264 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜑)
63, 5bitri 276 . . . . 5 ([𝑦 / 𝑥][𝑎 / 𝑦]𝜑𝜑)
76sbbii 2074 . . . 4 ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑥 / 𝑎]𝜑)
8 sbv 2091 . . . 4 ([𝑥 / 𝑎]𝜑𝜑)
97, 8bitri 276 . . 3 ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑𝜑)
109gen2 1790 . 2 𝑥𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑𝜑)
11 df-ich 43434 . 2 ([𝑥𝑦]𝜑 ↔ ∀𝑥𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑𝜑))
1210, 11mpbir 232 1 [𝑥𝑦]𝜑
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207  ∀wal 1528  Ⅎwnf 1777  [wsb 2062  [wich 43433 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-12 2169 This theorem depends on definitions:  df-bi 208  df-ex 1774  df-nf 1778  df-sb 2063  df-ich 43434 This theorem is referenced by:  ich2al  43456  ich2ex  43457
 Copyright terms: Public domain W3C validator