Theorem dedths2

Description: Generalization of dedths that is not useful unless we can separately prove |- A e. _V . (Contributed by NM, 13-Jun-2019)

	Ref	Expression
Hypothesis	dedths2.1	⊢ [ if ( [ 𝐴 / 𝑥 ] 𝜑 , 𝐴 , 𝐵 ) / 𝑥 ] 𝜓
Assertion	dedths2	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		dedths2.1	⊢ [ if ( [ 𝐴 / 𝑥 ] 𝜑 , 𝐴 , 𝐵 ) / 𝑥 ] 𝜓
2		dfsbcq	⊢ ( 𝐴 = if ( [ 𝐴 / 𝑥 ] 𝜑 , 𝐴 , 𝐵 ) → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ if ( [ 𝐴 / 𝑥 ] 𝜑 , 𝐴 , 𝐵 ) / 𝑥 ] 𝜓 ) )
3	2 1	dedth	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 )