Theorem elimhyps

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

	Ref	Expression
Hypothesis	elimhyps.1	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi$$
Assertion	elimhyps	$$⊢\underset{\dot{}}{[}if\left(\phi ,x,B\right)/x\underset{\dot{}}{]}\phi$$

Proof

Step	Hyp	Ref	Expression
1		elimhyps.1	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi$$
2		sbceq1a	$$⊢x=if\left(\phi ,x,B\right)\to \left(\phi \leftrightarrow \underset{\dot{}}{[}if\left(\phi ,x,B\right)/x\underset{\dot{}}{]}\phi \right)$$
3		dfsbcq	$$⊢B=if\left(\phi ,x,B\right)\to \left(\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}if\left(\phi ,x,B\right)/x\underset{\dot{}}{]}\phi \right)$$
4	2 3 1	elimhyp	$$⊢\underset{\dot{}}{[}if\left(\phi ,x,B\right)/x\underset{\dot{}}{]}\phi$$