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 )