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Theorem bnj1241 31206
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 2766 . . 3 (𝜓𝐴 = 𝐶)
32adantl 473 . 2 ((𝜑𝜓) → 𝐴 = 𝐶)
4 bnj1241.1 . . 3 (𝜑𝐴𝐵)
54adantr 472 . 2 ((𝜑𝜓) → 𝐴𝐵)
63, 5eqsstr3d 3781 1 ((𝜑𝜓) → 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wss 3715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-in 3722  df-ss 3729
This theorem is referenced by:  bnj1245  31410  bnj1311  31420
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