Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ax12 Structured version   Visualization version   GIF version

Theorem bj-ax12 33887
Description: Remove a DV condition from bj-ax12v 33886 (using core axioms only). (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ax12 𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
Distinct variable group:   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ax12
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bj-ax12v 33886 . . 3 𝑥(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 equequ2 2024 . . . . 5 (𝑦 = 𝑡 → (𝑥 = 𝑦𝑥 = 𝑡))
32imbi1d 343 . . . . . . 7 (𝑦 = 𝑡 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
43albidv 1912 . . . . . 6 (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑)))
54imbi2d 342 . . . . 5 (𝑦 = 𝑡 → ((𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))))
62, 5imbi12d 346 . . . 4 (𝑦 = 𝑡 → ((𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))) ↔ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))))
76albidv 1912 . . 3 (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))))
81, 7mpbii 234 . 2 (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))))
9 ax6ev 1963 . 2 𝑦 𝑦 = 𝑡
108, 9exlimiiv 1923 1 𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772
This theorem is referenced by:  bj-ax12ssb  33888  bj-sb56  33891
  Copyright terms: Public domain W3C validator