Theorem ceqsrex2v

Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005)

	Ref	Expression
Hypotheses	ceqsrex2v.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	ceqsrex2v.2	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
Assertion	ceqsrex2v	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐷 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ceqsrex2v.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		ceqsrex2v.2	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
3		anass	⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ ( 𝑥 = 𝐴 ∧ ( 𝑦 = 𝐵 ∧ 𝜑 ) ) )
4	3	rexbii	⊢ ( ∃ 𝑦 ∈ 𝐷 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ ∃ 𝑦 ∈ 𝐷 ( 𝑥 = 𝐴 ∧ ( 𝑦 = 𝐵 ∧ 𝜑 ) ) )
5		r19.42v	⊢ ( ∃ 𝑦 ∈ 𝐷 ( 𝑥 = 𝐴 ∧ ( 𝑦 = 𝐵 ∧ 𝜑 ) ) ↔ ( 𝑥 = 𝐴 ∧ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜑 ) ) )
6	4 5	bitri	⊢ ( ∃ 𝑦 ∈ 𝐷 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ ( 𝑥 = 𝐴 ∧ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜑 ) ) )
7	6	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐷 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ ∃ 𝑥 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜑 ) ) )
8	1	anbi2d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑦 = 𝐵 ∧ 𝜑 ) ↔ ( 𝑦 = 𝐵 ∧ 𝜓 ) ) )
9	8	rexbidv	⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜑 ) ↔ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜓 ) ) )
10	9	ceqsrexv	⊢ ( 𝐴 ∈ 𝐶 → ( ∃ 𝑥 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜑 ) ) ↔ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜓 ) ) )
11	7 10	bitrid	⊢ ( 𝐴 ∈ 𝐶 → ( ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐷 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜓 ) ) )
12	2	ceqsrexv	⊢ ( 𝐵 ∈ 𝐷 → ( ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜓 ) ↔ 𝜒 ) )
13	11 12	sylan9bb	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐷 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ 𝜒 ) )