Theorem elimhyps2

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

	Ref	Expression
Hypothesis	elimhyps2.1	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi$$
Assertion	elimhyps2	$$⊢\underset{\dot{}}{[}if\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi ,A,B\right)/x\underset{\dot{}}{]}\phi$$

Proof

Step	Hyp	Ref	Expression
1		elimhyps2.1	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi$$
2		dfsbcq	$$⊢A=if\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi ,A,B\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}if\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi ,A,B\right)/x\underset{\dot{}}{]}\phi \right)$$
3		dfsbcq	$$⊢B=if\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi ,A,B\right)\to \left(\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}if\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi ,A,B\right)/x\underset{\dot{}}{]}\phi \right)$$
4	2 3 1	elimhyp	$$⊢\underset{\dot{}}{[}if\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi ,A,B\right)/x\underset{\dot{}}{]}\phi$$