 Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ssblem2 Structured version   Visualization version   GIF version

Theorem bj-ssblem2 32265
 Description: An instance of ax-11 2036 proved without it. The converse may not be provable without ax-11 2036 (since using alcomiw 1973 would require a DV on 𝜑, 𝑥, which defeats the purpose). (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-ssblem2 (∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
Distinct variable groups:   𝑥,𝑦,𝑡   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ssblem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equequ1 1954 . . 3 (𝑦 = 𝑧 → (𝑦 = 𝑡𝑧 = 𝑡))
2 equequ2 1955 . . . 4 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
32imbi1d 331 . . 3 (𝑦 = 𝑧 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑧𝜑)))
41, 3imbi12d 334 . 2 (𝑦 = 𝑧 → ((𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) ↔ (𝑧 = 𝑡 → (𝑥 = 𝑧𝜑))))
54alcomiw 1973 1 (∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1478 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 1841  ax-6 1890  ax-7 1937 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by:  bj-ssb1a  32266
 Copyright terms: Public domain W3C validator