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Theorem bnj1241 31724
 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 2785 . . 3 (𝜓𝐴 = 𝐶)
32adantl 474 . 2 ((𝜑𝜓) → 𝐴 = 𝐶)
4 bnj1241.1 . . 3 (𝜑𝐴𝐵)
54adantr 473 . 2 ((𝜑𝜓) → 𝐴𝐵)
63, 5eqsstr3d 3897 1 ((𝜑𝜓) → 𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ⊆ wss 3830 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-in 3837  df-ss 3844 This theorem is referenced by:  bnj1245  31928  bnj1311  31938
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