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Metamath Proof Explorer

Theorem rmob2

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 )
Assertion rmob2 ( 𝜑 → ( 𝑥 = 𝐵 ↔ 𝜒 ) )

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 df-rmo ⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
7 3 6 sylib ⊢ ( 𝜑 → ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
8 eleq1 ⊢ ( 𝑥 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴 ) )
9 8 1 anbi12d ⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝐵 ∈ 𝐴 ∧ 𝜒 ) ) )
10 9 mob2 ⊢ ( ( 𝐵 ∈ 𝐴 ∧ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) → ( 𝑥 = 𝐵 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝜒 ) ) )
11 2 7 4 5 10 syl112anc ⊢ ( 𝜑 → ( 𝑥 = 𝐵 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝜒 ) ) )
12 2 11 mpbirand ⊢ ( 𝜑 → ( 𝑥 = 𝐵 ↔ 𝜒 ) )