Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ssbequ1 Structured version   Visualization version   GIF version

Theorem bj-ssbequ1 32997
Description: This uses ax-12 2206 with a direct reference to ax12v 2207. Therefore, compared to bj-ax12 32988, there is a hidden use of sp 2210. Note that with ax-12 2206, it can be proved with dv condition on 𝑥, 𝑡. See sbequ1 2269. (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-ssbequ1 (𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑))

Proof of Theorem bj-ssbequ1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 equtr2 2115 . . . . . . . 8 ((𝑦 = 𝑡𝑥 = 𝑡) → 𝑦 = 𝑥)
21equcomd 2107 . . . . . . 7 ((𝑦 = 𝑡𝑥 = 𝑡) → 𝑥 = 𝑦)
3 ax12v 2207 . . . . . . 7 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
42, 3syl 17 . . . . . 6 ((𝑦 = 𝑡𝑥 = 𝑡) → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
54expimpd 442 . . . . 5 (𝑦 = 𝑡 → ((𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
65com12 32 . . . 4 ((𝑥 = 𝑡𝜑) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
76alrimiv 2010 . . 3 ((𝑥 = 𝑡𝜑) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
87ex 398 . 2 (𝑥 = 𝑡 → (𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑))))
9 df-ssb 32974 . 2 ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
108, 9syl6ibr 243 1 (𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1632  [wssb 32973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-12 2206
This theorem depends on definitions:  df-bi 198  df-an 384  df-ex 1856  df-ssb 32974
This theorem is referenced by:  bj-ssbid1  33000
  Copyright terms: Public domain W3C validator