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Theorem bnj1541 31443
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 314 . . 3 ¬ (𝜓𝐴𝐵)
43imnani 390 . 2 (𝜓 → ¬ 𝐴𝐵)
5 nne 2975 . 2 𝐴𝐵𝐴 = 𝐵)
64, 5sylib 210 1 (𝜓𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wne 2971
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 199  df-an 386  df-ne 2972
This theorem is referenced by:  bnj1312  31643  bnj1523  31656
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