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	$$⊢\underset{\dot{}}{[}if\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi ,A,B\right)/x\underset{\dot{}}{]}\psi$$
Assertion	dedths2	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi$$

Proof

Step	Hyp	Ref	Expression
1		dedths2.1	$$⊢\underset{\dot{}}{[}if\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi ,A,B\right)/x\underset{\dot{}}{]}\psi$$
2		dfsbcq	$$⊢A=if\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi ,A,B\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \underset{\dot{}}{[}if\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi ,A,B\right)/x\underset{\dot{}}{]}\psi \right)$$
3	2 1	dedth	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi$$