Theorem dedths

Description: A version of weak deduction theorem dedth using explicit substitution. (Contributed by NM, 15-Jun-2019)

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

Proof

Step	Hyp	Ref	Expression
1		dedths.1	⊢ [ if ( 𝜑 , 𝑥 , 𝐵 ) / 𝑥 ] 𝜓
2		dfsbcq	⊢ ( 𝑥 = if ( [ 𝑥 / 𝑥 ] 𝜑 , 𝑥 , 𝐵 ) → ( [ 𝑥 / 𝑥 ] 𝜓 ↔ [ if ( [ 𝑥 / 𝑥 ] 𝜑 , 𝑥 , 𝐵 ) / 𝑥 ] 𝜓 ) )
3		sbcid	⊢ ( [ 𝑥 / 𝑥 ] 𝜑 ↔ 𝜑 )
4		ifbi	⊢ ( ( [ 𝑥 / 𝑥 ] 𝜑 ↔ 𝜑 ) → if ( [ 𝑥 / 𝑥 ] 𝜑 , 𝑥 , 𝐵 ) = if ( 𝜑 , 𝑥 , 𝐵 ) )
5		dfsbcq	⊢ ( if ( [ 𝑥 / 𝑥 ] 𝜑 , 𝑥 , 𝐵 ) = if ( 𝜑 , 𝑥 , 𝐵 ) → ( [ if ( [ 𝑥 / 𝑥 ] 𝜑 , 𝑥 , 𝐵 ) / 𝑥 ] 𝜓 ↔ [ if ( 𝜑 , 𝑥 , 𝐵 ) / 𝑥 ] 𝜓 ) )
6	3 4 5	mp2b	⊢ ( [ if ( [ 𝑥 / 𝑥 ] 𝜑 , 𝑥 , 𝐵 ) / 𝑥 ] 𝜓 ↔ [ if ( 𝜑 , 𝑥 , 𝐵 ) / 𝑥 ] 𝜓 )
7	1 6	mpbir	⊢ [ if ( [ 𝑥 / 𝑥 ] 𝜑 , 𝑥 , 𝐵 ) / 𝑥 ] 𝜓
8	2 7	dedth	⊢ ( [ 𝑥 / 𝑥 ] 𝜑 → [ 𝑥 / 𝑥 ] 𝜓 )
9		sbcid	⊢ ( [ 𝑥 / 𝑥 ] 𝜓 ↔ 𝜓 )
10	8 3 9	3imtr3i	⊢ ( 𝜑 → 𝜓 )