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Metamath Proof Explorer

Theorem rmoi

Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012) (Revised by NM, 16-Jun-2017)

Ref Expression
Hypotheses rmoi.b ⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
rmoi.c ⊢ ( 𝑥 = 𝐶 → ( 𝜑 ↔ 𝜒 ) )
Assertion rmoi ( ( ∃* 𝑥 ∈ 𝐴 𝜑 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝜓 ) ∧ ( 𝐶 ∈ 𝐴 ∧ 𝜒 ) ) → 𝐵 = 𝐶 )

Proof

Step Hyp Ref Expression
1 rmoi.b ⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
2 rmoi.c ⊢ ( 𝑥 = 𝐶 → ( 𝜑 ↔ 𝜒 ) )
3 1 2 rmob ⊢ ( ( ∃* 𝑥 ∈ 𝐴 𝜑 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝜓 ) ) → ( 𝐵 = 𝐶 ↔ ( 𝐶 ∈ 𝐴 ∧ 𝜒 ) ) )
4 3 biimp3ar ⊢ ( ( ∃* 𝑥 ∈ 𝐴 𝜑 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝜓 ) ∧ ( 𝐶 ∈ 𝐴 ∧ 𝜒 ) ) → 𝐵 = 𝐶 )