Theorem csbied

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014) (Revised by Mario Carneiro, 13-Oct-2016) Reduce axiom usage. (Revised by Gino Giotto, 15-Oct-2024)

	Ref	Expression
Hypotheses	csbied.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	csbied.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 = 𝐶 )
Assertion	csbied	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		csbied.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		csbied.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 = 𝐶 )
3		df-csb	⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 }
4	2	eleq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶 ) )
5	1 4	sbcied	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶 ) )
6	5	alrimiv	⊢ ( 𝜑 → ∀ 𝑧 ( [ 𝐴 / 𝑥 ] 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶 ) )
7		df-clab	⊢ ( 𝑧 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 } ↔ [ 𝑧 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 )
8		eleq1w	⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵 ) )
9	8	sbcbidv	⊢ ( 𝑦 = 𝑧 → ( [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 ↔ [ 𝐴 / 𝑥 ] 𝑧 ∈ 𝐵 ) )
10	9	sbievw	⊢ ( [ 𝑧 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 ↔ [ 𝐴 / 𝑥 ] 𝑧 ∈ 𝐵 )
11	7 10	bitr2i	⊢ ( [ 𝐴 / 𝑥 ] 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 } )
12	11	bibi1i	⊢ ( ( [ 𝐴 / 𝑥 ] 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶 ) ↔ ( 𝑧 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 } ↔ 𝑧 ∈ 𝐶 ) )
13	12	biimpi	⊢ ( ( [ 𝐴 / 𝑥 ] 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶 ) → ( 𝑧 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 } ↔ 𝑧 ∈ 𝐶 ) )
14	6 13	sylg	⊢ ( 𝜑 → ∀ 𝑧 ( 𝑧 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 } ↔ 𝑧 ∈ 𝐶 ) )
15		dfcleq	⊢ ( { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 } = 𝐶 ↔ ∀ 𝑧 ( 𝑧 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 } ↔ 𝑧 ∈ 𝐶 ) )
16	14 15	sylibr	⊢ ( 𝜑 → { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 } = 𝐶 )
17	3 16	eqtrid	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 )