Theorem sbc5

Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993) (Revised by Mario Carneiro, 12-Oct-2016)

	Ref	Expression
Assertion	sbc5	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sbcex	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → 𝐴 ∈ V )
2		exsimpl	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) → ∃ 𝑥 𝑥 = 𝐴 )
3		isset	⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )
4	2 3	sylibr	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) → 𝐴 ∈ V )
5		dfsbcq2	⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )
6		eqeq2	⊢ ( 𝑦 = 𝐴 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝐴 ) )
7	6	anbi1d	⊢ ( 𝑦 = 𝐴 → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ( 𝑥 = 𝐴 ∧ 𝜑 ) ) )
8	7	exbidv	⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ) )
9		sb5	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )
10	5 8 9	vtoclbg	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ) )
11	1 4 10	pm5.21nii	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) )