Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ssbid1ALT Structured version   Visualization version   GIF version

Theorem bj-ssbid1ALT 33884
Description: Alternate proof of bj-ssbid1 33883, not using sbequ1 2242. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-ssbid1ALT (𝜑 → [𝑥 / 𝑥]𝜑)

Proof of Theorem bj-ssbid1ALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax12v 2170 . . . . 5 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
21equcoms 2020 . . . 4 (𝑦 = 𝑥 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
32com12 32 . . 3 (𝜑 → (𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
43alrimiv 1921 . 2 (𝜑 → ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
5 df-sb 2063 . 2 ([𝑥 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
64, 5sylibr 235 1 (𝜑 → [𝑥 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528  [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-12 2169
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-sb 2063
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator