Theorem ralab

Description: Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010) Reduce axiom usage. (Revised by Gino Giotto, 2-Nov-2024)

	Ref	Expression
Hypothesis	ralab.1	⊢ ( 𝑦 = 𝑥 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ralab	⊢ ( ∀ 𝑥 ∈ { 𝑦 ∣ 𝜑 } 𝜒 ↔ ∀ 𝑥 ( 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralab.1	⊢ ( 𝑦 = 𝑥 → ( 𝜑 ↔ 𝜓 ) )
2		df-ral	⊢ ( ∀ 𝑥 ∈ { 𝑦 ∣ 𝜑 } 𝜒 ↔ ∀ 𝑥 ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } → 𝜒 ) )
3		df-clab	⊢ ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } ↔ [ 𝑥 / 𝑦 ] 𝜑 )
4	1	sbievw	⊢ ( [ 𝑥 / 𝑦 ] 𝜑 ↔ 𝜓 )
5	3 4	bitri	⊢ ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } ↔ 𝜓 )
6	5	imbi1i	⊢ ( ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } → 𝜒 ) ↔ ( 𝜓 → 𝜒 ) )
7		biid	⊢ ( ( 𝜓 → 𝜒 ) ↔ ( 𝜓 → 𝜒 ) )
8	6 7	bitri	⊢ ( ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } → 𝜒 ) ↔ ( 𝜓 → 𝜒 ) )
9	8	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } → 𝜒 ) ↔ ∀ 𝑥 ( 𝜓 → 𝜒 ) )
10	2 9	bitri	⊢ ( ∀ 𝑥 ∈ { 𝑦 ∣ 𝜑 } 𝜒 ↔ ∀ 𝑥 ( 𝜓 → 𝜒 ) )