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Metamath Proof Explorer

Theorem rmoi2

Description: Consequence of "restricted at most one". (Contributed by Thierry Arnoux, 9-Dec-2019)

Ref Expression
Hypotheses rmoi2.1 ⊢ ( 𝑥 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
rmoi2.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
rmoi2.3 ⊢ ( 𝜑 → ∃* 𝑥 ∈ 𝐴 𝜓 )
rmoi2.4 ⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
rmoi2.5 ⊢ ( 𝜑 → 𝜓 )
rmoi2.6 ⊢ ( 𝜑 → 𝜒 )
Assertion rmoi2 ( 𝜑 → 𝑥 = 𝐵 )

Proof

Step Hyp Ref Expression
1 rmoi2.1 ⊢ ( 𝑥 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
2 rmoi2.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
3 rmoi2.3 ⊢ ( 𝜑 → ∃* 𝑥 ∈ 𝐴 𝜓 )
4 rmoi2.4 ⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
5 rmoi2.5 ⊢ ( 𝜑 → 𝜓 )
6 rmoi2.6 ⊢ ( 𝜑 → 𝜒 )
7 1 2 3 4 5 rmob2 ⊢ ( 𝜑 → ( 𝑥 = 𝐵 ↔ 𝜒 ) )
8 6 7 mpbird ⊢ ( 𝜑 → 𝑥 = 𝐵 )