Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ssbequ2 Structured version   Visualization version   GIF version

Theorem bj-ssbequ2 32980
Description: Note that ax-12 2203 is used only via sp 2207. See sbequ2 2051 and stdpc7 2114. (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-ssbequ2 (𝑥 = 𝑡 → ([𝑡/𝑥]b𝜑𝜑))

Proof of Theorem bj-ssbequ2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-ssb 32958 . . 3 ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 sp 2207 . . . . . 6 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
32imim2i 16 . . . . 5 ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
43alimi 1887 . . . 4 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
5 pm3.31 436 . . . . 5 ((𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ((𝑦 = 𝑡𝑥 = 𝑦) → 𝜑))
65alimi 1887 . . . 4 (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦((𝑦 = 𝑡𝑥 = 𝑦) → 𝜑))
7 19.23v 2023 . . . . 5 (∀𝑦((𝑦 = 𝑡𝑥 = 𝑦) → 𝜑) ↔ (∃𝑦(𝑦 = 𝑡𝑥 = 𝑦) → 𝜑))
8 equvinva 2117 . . . . . . 7 (𝑥 = 𝑡 → ∃𝑦(𝑥 = 𝑦𝑡 = 𝑦))
9 biid 251 . . . . . . . . . . . 12 (𝑥 = 𝑦𝑥 = 𝑦)
10 equcom 2103 . . . . . . . . . . . 12 (𝑡 = 𝑦𝑦 = 𝑡)
119, 10anbi12ci 613 . . . . . . . . . . 11 ((𝑥 = 𝑦𝑡 = 𝑦) ↔ (𝑦 = 𝑡𝑥 = 𝑦))
1211biimpi 206 . . . . . . . . . 10 ((𝑥 = 𝑦𝑡 = 𝑦) → (𝑦 = 𝑡𝑥 = 𝑦))
1312eximi 1910 . . . . . . . . 9 (∃𝑦(𝑥 = 𝑦𝑡 = 𝑦) → ∃𝑦(𝑦 = 𝑡𝑥 = 𝑦))
14 pm3.35 804 . . . . . . . . 9 ((∃𝑦(𝑦 = 𝑡𝑥 = 𝑦) ∧ (∃𝑦(𝑦 = 𝑡𝑥 = 𝑦) → 𝜑)) → 𝜑)
1513, 14sylan 569 . . . . . . . 8 ((∃𝑦(𝑥 = 𝑦𝑡 = 𝑦) ∧ (∃𝑦(𝑦 = 𝑡𝑥 = 𝑦) → 𝜑)) → 𝜑)
1615ancoms 455 . . . . . . 7 (((∃𝑦(𝑦 = 𝑡𝑥 = 𝑦) → 𝜑) ∧ ∃𝑦(𝑥 = 𝑦𝑡 = 𝑦)) → 𝜑)
178, 16sylan2 580 . . . . . 6 (((∃𝑦(𝑦 = 𝑡𝑥 = 𝑦) → 𝜑) ∧ 𝑥 = 𝑡) → 𝜑)
1817ex 397 . . . . 5 ((∃𝑦(𝑦 = 𝑡𝑥 = 𝑦) → 𝜑) → (𝑥 = 𝑡𝜑))
197, 18sylbi 207 . . . 4 (∀𝑦((𝑦 = 𝑡𝑥 = 𝑦) → 𝜑) → (𝑥 = 𝑡𝜑))
204, 6, 193syl 18 . . 3 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑥 = 𝑡𝜑))
211, 20sylbi 207 . 2 ([𝑡/𝑥]b𝜑 → (𝑥 = 𝑡𝜑))
2221com12 32 1 (𝑥 = 𝑡 → ([𝑡/𝑥]b𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1629  wex 1852  [wssb 32957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-ssb 32958
This theorem is referenced by:  bj-ssbid2  32982
  Copyright terms: Public domain W3C validator