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Theorem bnj1241 32086
 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 2827 . . 3 (𝜓𝐴 = 𝐶)
32adantl 485 . 2 ((𝜑𝜓) → 𝐴 = 𝐶)
4 bnj1241.1 . . 3 (𝜑𝐴𝐵)
54adantr 484 . 2 ((𝜑𝜓) → 𝐴𝐵)
63, 5eqsstrrd 3982 1 ((𝜑𝜓) → 𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ⊆ wss 3910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473  df-in 3917  df-ss 3927 This theorem is referenced by:  bnj1245  32293  bnj1311  32303
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