Metamath Proof Explorer


Theorem bnj1541

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1541.1 ( 𝜑 ↔ ( 𝜓𝐴𝐵 ) )
bnj1541.2 ¬ 𝜑
Assertion bnj1541 ( 𝜓𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 bnj1541.1 ( 𝜑 ↔ ( 𝜓𝐴𝐵 ) )
2 bnj1541.2 ¬ 𝜑
3 2 1 mtbi ¬ ( 𝜓𝐴𝐵 )
4 3 imnani ( 𝜓 → ¬ 𝐴𝐵 )
5 nne ( ¬ 𝐴𝐵𝐴 = 𝐵 )
6 4 5 sylib ( 𝜓𝐴 = 𝐵 )