New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  sbid2 GIF version

Theorem sbid2 2084
 Description: An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sbid2.1 xφ
Assertion
Ref Expression
sbid2 ([y / x][x / y]φφ)

Proof of Theorem sbid2
StepHypRef Expression
1 sbco 2083 . 2 ([y / x][x / y]φ ↔ [y / x]φ)
2 sbid2.1 . . 3 xφ
32sbf 2026 . 2 ([y / x]φφ)
41, 3bitri 240 1 ([y / x][x / y]φφ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  Ⅎwnf 1544  [wsb 1648 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sbco2  2086  sb5rf  2090  sb6rf  2091  sbid2v  2123
 Copyright terms: Public domain W3C validator