Theorem cpcoll2d

Description: cpcolld with an extra existential quantifier. (Contributed by Rohan Ridenour, 12-Aug-2023)

	Ref	Expression
Hypotheses	cpcoll2d.1	⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
	cpcoll2d.2	⊢ ( 𝜑 → ∃ 𝑦 𝑥 𝐹 𝑦 )
Assertion	cpcoll2d	⊢ ( 𝜑 → ∃ 𝑦 ∈ ( 𝐹 Coll 𝐴 ) 𝑥 𝐹 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		cpcoll2d.1	⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
2		cpcoll2d.2	⊢ ( 𝜑 → ∃ 𝑦 𝑥 𝐹 𝑦 )
3		breq2	⊢ ( 𝑎 = 𝑦 → ( 𝑥 𝐹 𝑎 ↔ 𝑥 𝐹 𝑦 ) )
4	3	cbvexvw	⊢ ( ∃ 𝑎 𝑥 𝐹 𝑎 ↔ ∃ 𝑦 𝑥 𝐹 𝑦 )
5	2 4	sylibr	⊢ ( 𝜑 → ∃ 𝑎 𝑥 𝐹 𝑎 )
6	1	adantr	⊢ ( ( 𝜑 ∧ 𝑥 𝐹 𝑎 ) → 𝑥 ∈ 𝐴 )
7		simpr	⊢ ( ( 𝜑 ∧ 𝑥 𝐹 𝑎 ) → 𝑥 𝐹 𝑎 )
8	6 7	cpcolld	⊢ ( ( 𝜑 ∧ 𝑥 𝐹 𝑎 ) → ∃ 𝑎 ∈ ( 𝐹 Coll 𝐴 ) 𝑥 𝐹 𝑎 )
9	3	cbvrexvw	⊢ ( ∃ 𝑎 ∈ ( 𝐹 Coll 𝐴 ) 𝑥 𝐹 𝑎 ↔ ∃ 𝑦 ∈ ( 𝐹 Coll 𝐴 ) 𝑥 𝐹 𝑦 )
10	8 9	sylib	⊢ ( ( 𝜑 ∧ 𝑥 𝐹 𝑎 ) → ∃ 𝑦 ∈ ( 𝐹 Coll 𝐴 ) 𝑥 𝐹 𝑦 )
11	5 10	exlimddv	⊢ ( 𝜑 → ∃ 𝑦 ∈ ( 𝐹 Coll 𝐴 ) 𝑥 𝐹 𝑦 )