Theorem elimhyps

Description: A version of elimhyp using explicit substitution. (Contributed by NM, 15-Jun-2019)

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

Proof

Step	Hyp	Ref	Expression
1		elimhyps.1	⊢ [ 𝐵 / 𝑥 ] 𝜑
2		sbceq1a	⊢ ( 𝑥 = if ( 𝜑 , 𝑥 , 𝐵 ) → ( 𝜑 ↔ [ if ( 𝜑 , 𝑥 , 𝐵 ) / 𝑥 ] 𝜑 ) )
3		dfsbcq	⊢ ( 𝐵 = if ( 𝜑 , 𝑥 , 𝐵 ) → ( [ 𝐵 / 𝑥 ] 𝜑 ↔ [ if ( 𝜑 , 𝑥 , 𝐵 ) / 𝑥 ] 𝜑 ) )
4	2 3 1	elimhyp	⊢ [ if ( 𝜑 , 𝑥 , 𝐵 ) / 𝑥 ] 𝜑