Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ax12ssb Structured version   Visualization version   GIF version

Theorem bj-ax12ssb 31824
 Description: The axiom bj-ax12 31823 expressed using substitution. (Contributed by BJ, 26-Dec-2020.)
Assertion
Ref Expression
bj-ax12ssb [𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑)
Distinct variable group:   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ax12ssb
StepHypRef Expression
1 bj-ax12 31823 . . 3 𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
2 bj-ssb1 31822 . . . . . 6 ([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))
32imbi2i 325 . . . . 5 ((𝜑 → [𝑡/𝑥]b𝜑) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
43imbi2i 325 . . . 4 ((𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑)) ↔ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))))
54albii 1737 . . 3 (∀𝑥(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))))
61, 5mpbir 220 . 2 𝑥(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑))
7 bj-ssb1 31822 . 2 ([𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑)))
86, 7mpbir 220 1 [𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  [wssb 31808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-11 2021  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ssb 31809 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator