Theorem reurexprg

Description: Convert a restricted existential uniqueness over a pair to a restricted existential quantification and an implication . (Contributed by AV, 3-Apr-2023)

	Ref	Expression
Hypotheses	reuprg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	reuprg.2	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
Assertion	reurexprg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃! 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ∧ ( ( 𝜒 ∧ 𝜓 ) → 𝐴 = 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		reuprg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		reuprg.2	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
3	1 2	reuprg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃! 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( ( 𝜓 ∨ 𝜒 ) ∧ ( ( 𝜒 ∧ 𝜓 ) → 𝐴 = 𝐵 ) ) ) )
4	1 2	rexprg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∨ 𝜒 ) ) )
5	4	bicomd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝜓 ∨ 𝜒 ) ↔ ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ) )
6	5	anbi1d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( ( 𝜓 ∨ 𝜒 ) ∧ ( ( 𝜒 ∧ 𝜓 ) → 𝐴 = 𝐵 ) ) ↔ ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ∧ ( ( 𝜒 ∧ 𝜓 ) → 𝐴 = 𝐵 ) ) ) )
7	3 6	bitrd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃! 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ∧ ( ( 𝜒 ∧ 𝜓 ) → 𝐴 = 𝐵 ) ) ) )