Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1241 Structured version   Visualization version   GIF version

Theorem bnj1241 32787
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1241.1 (𝜑𝐴𝐵)
bnj1241.2 (𝜓𝐶 = 𝐴)
Assertion
Ref Expression
bnj1241 ((𝜑𝜓) → 𝐶𝐵)

Proof of Theorem bnj1241
StepHypRef Expression
1 bnj1241.2 . . . 4 (𝜓𝐶 = 𝐴)
21eqcomd 2744 . . 3 (𝜓𝐴 = 𝐶)
32adantl 482 . 2 ((𝜑𝜓) → 𝐴 = 𝐶)
4 bnj1241.1 . . 3 (𝜑𝐴𝐵)
54adantr 481 . 2 ((𝜑𝜓) → 𝐴𝐵)
63, 5eqsstrrd 3960 1 ((𝜑𝜓) → 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wss 3887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904
This theorem is referenced by:  bnj1245  32994  bnj1311  33004
  Copyright terms: Public domain W3C validator