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 30132
 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 2616 . . 3 (𝜓𝐴 = 𝐶)
32adantl 481 . 2 ((𝜑𝜓) → 𝐴 = 𝐶)
4 bnj1241.1 . . 3 (𝜑𝐴𝐵)
54adantr 480 . 2 ((𝜑𝜓) → 𝐴𝐵)
63, 5eqsstr3d 3603 1 ((𝜑𝜓) → 𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ⊆ wss 3540 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554 This theorem is referenced by:  bnj1245  30336  bnj1311  30346
 Copyright terms: Public domain W3C validator