Theorem cbvexfo

Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997)

	Ref	Expression
Hypothesis	cbvfo.1	⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvexfo	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvfo.1	⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2	1	notbid	⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑦 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3	2	cbvfo	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 ¬ 𝜓 ) )
4	3	notbid	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( ¬ ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀ 𝑦 ∈ 𝐵 ¬ 𝜓 ) )
5		dfrex2	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 )
6		dfrex2	⊢ ( ∃ 𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∀ 𝑦 ∈ 𝐵 ¬ 𝜓 )
7	4 5 6	3bitr4g	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 𝜓 ) )