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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 ↔ ( 𝜓 ∧ 𝐴 ≠ 𝐵 ) )
bnj1541.2 ⊢ ¬ 𝜑
Assertion bnj1541 ( 𝜓 → 𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 bnj1541.1 ⊢ ( 𝜑 ↔ ( 𝜓 ∧ 𝐴 ≠ 𝐵 ) )
2 bnj1541.2 ⊢ ¬ 𝜑
3 2 1 mtbi ⊢ ¬ ( 𝜓 ∧ 𝐴 ≠ 𝐵 )
4 3 imnani ⊢ ( 𝜓 → ¬ 𝐴 ≠ 𝐵 )
5 nne ⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )
6 4 5 sylib ⊢ ( 𝜓 → 𝐴 = 𝐵 )