Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ssbid2ALT Structured version   Visualization version   GIF version

Theorem bj-ssbid2ALT 33996
 Description: Alternate proof of bj-ssbid2 33995, not using sbequ2 2246. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-ssbid2ALT ([𝑥 / 𝑥]𝜑𝜑)

Proof of Theorem bj-ssbid2ALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-sb 2066 . 2 ([𝑥 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 sp 2178 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
32imim2i 16 . . . 4 ((𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)))
43alimi 1808 . . 3 (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑦(𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)))
5 pm2.21 123 . . . . . 6 𝑦 = 𝑥 → (𝑦 = 𝑥𝜑))
6 equcomi 2020 . . . . . . 7 (𝑦 = 𝑥𝑥 = 𝑦)
76imim1i 63 . . . . . 6 ((𝑥 = 𝑦𝜑) → (𝑦 = 𝑥𝜑))
85, 7ja 188 . . . . 5 ((𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)) → (𝑦 = 𝑥𝜑))
98alimi 1808 . . . 4 (∀𝑦(𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)) → ∀𝑦(𝑦 = 𝑥𝜑))
10 ax6ev 1968 . . . 4 𝑦 𝑦 = 𝑥
11 19.23v 1939 . . . . 5 (∀𝑦(𝑦 = 𝑥𝜑) ↔ (∃𝑦 𝑦 = 𝑥𝜑))
1211biimpi 218 . . . 4 (∀𝑦(𝑦 = 𝑥𝜑) → (∃𝑦 𝑦 = 𝑥𝜑))
139, 10, 12mpisyl 21 . . 3 (∀𝑦(𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)) → 𝜑)
144, 13syl 17 . 2 (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → 𝜑)
151, 14sylbi 219 1 ([𝑥 / 𝑥]𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1531  ∃wex 1776  [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-12 2173 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-sb 2066 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator