MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb6x Structured version   Visualization version   GIF version

Theorem sb6x 2543
Description: Equivalence involving substitution for a variable not free. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sb6x.1 𝑥𝜑
Assertion
Ref Expression
sb6x ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem sb6x
StepHypRef Expression
1 sb6x.1 . . 3 𝑥𝜑
21sbf 2539 . 2 ([𝑦 / 𝑥]𝜑𝜑)
3 biidd 253 . . 3 (𝑥 = 𝑦 → (𝜑𝜑))
41, 3equsal 2458 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)
52, 4bitr4i 269 1 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1635  wnf 1863  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-12 2214  ax-13 2420
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ex 1860  df-nf 1864  df-sb 2061
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator