Theorem rexxfrd

Description: Transfer existential quantification from a variable x to another variable y contained in expression A . (Contributed by FL, 10-Apr-2007) (Revised by Mario Carneiro, 15-Aug-2014)

	Ref	Expression
Hypotheses	ralxfrd.1	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
	ralxfrd.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
	ralxfrd.3	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	rexxfrd	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 𝜓 ↔ ∃ 𝑦 ∈ 𝐶 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralxfrd.1	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
2		ralxfrd.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
3		ralxfrd.3	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
4	3	notbid	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( ¬ 𝜓 ↔ ¬ 𝜒 ) )
5	1 2 4	ralxfrd	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 ¬ 𝜓 ↔ ∀ 𝑦 ∈ 𝐶 ¬ 𝜒 ) )
6	5	notbid	⊢ ( 𝜑 → ( ¬ ∀ 𝑥 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∀ 𝑦 ∈ 𝐶 ¬ 𝜒 ) )
7		dfrex2	⊢ ( ∃ 𝑥 ∈ 𝐵 𝜓 ↔ ¬ ∀ 𝑥 ∈ 𝐵 ¬ 𝜓 )
8		dfrex2	⊢ ( ∃ 𝑦 ∈ 𝐶 𝜒 ↔ ¬ ∀ 𝑦 ∈ 𝐶 ¬ 𝜒 )
9	6 7 8	3bitr4g	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 𝜓 ↔ ∃ 𝑦 ∈ 𝐶 𝜒 ) )