Theorem sbcied

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014) Avoid ax-10 , ax-12 . (Revised by GG, 12-Oct-2024)

	Ref	Expression
Hypotheses	sbcied.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	sbcied.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	sbcied	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		sbcied.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		sbcied.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3		df-sbc	⊢ ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝐴 ∈ { 𝑥 ∣ 𝜓 } )
4	1 2	elabd3	⊢ ( 𝜑 → ( 𝐴 ∈ { 𝑥 ∣ 𝜓 } ↔ 𝜒 ) )
5	3 4	bitrid	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) )