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Theorem bnj1541 31052
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1541.1 (𝜑 ↔ (𝜓𝐴𝐵))
bnj1541.2 ¬ 𝜑
Assertion
Ref Expression
bnj1541 (𝜓𝐴 = 𝐵)

Proof of Theorem bnj1541
StepHypRef Expression
1 bnj1541.2 . . . 4 ¬ 𝜑
2 bnj1541.1 . . . 4 (𝜑 ↔ (𝜓𝐴𝐵))
31, 2mtbi 311 . . 3 ¬ (𝜓𝐴𝐵)
43imnani 438 . 2 (𝜓 → ¬ 𝐴𝐵)
5 nne 2827 . 2 𝐴𝐵𝐴 = 𝐵)
64, 5sylib 208 1 (𝜓𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ≠ wne 2823 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 197  df-an 385  df-ne 2824 This theorem is referenced by:  bnj1312  31252  bnj1523  31265
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