Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege60b Structured version   Visualization version   GIF version

Theorem frege60b 37720
Description: Swap antecedents of ax-frege58b 37716. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege60b (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))

Proof of Theorem frege60b
StepHypRef Expression
1 ax-frege58b 37716 . . 3 (∀𝑥(𝜑 → (𝜓𝜒)) → [𝑦 / 𝑥](𝜑 → (𝜓𝜒)))
2 sbim 2394 . . . 4 ([𝑦 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓𝜒)))
3 sbim 2394 . . . . 5 ([𝑦 / 𝑥](𝜓𝜒) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
43imbi2i 326 . . . 4 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
52, 4bitri 264 . . 3 ([𝑦 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
61, 5sylib 208 . 2 (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
7 frege12 37628 . 2 ((∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))) → (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒))))
86, 7ax-mp 5 1 (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245  ax-frege1 37605  ax-frege2 37606  ax-frege8 37624  ax-frege58b 37716
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator