Theorem keephyp

Description: Transform a hypothesis ps that we want to keep (but contains the same class variable A used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999)

	Ref	Expression
Hypotheses	keephyp.1	⊢ ( 𝐴 = if ( 𝜑 , 𝐴 , 𝐵 ) → ( 𝜓 ↔ 𝜃 ) )
	keephyp.2	⊢ ( 𝐵 = if ( 𝜑 , 𝐴 , 𝐵 ) → ( 𝜒 ↔ 𝜃 ) )
	keephyp.3	⊢ 𝜓
	keephyp.4	⊢ 𝜒
Assertion	keephyp	⊢ 𝜃

Proof

Step	Hyp	Ref	Expression
1		keephyp.1	⊢ ( 𝐴 = if ( 𝜑 , 𝐴 , 𝐵 ) → ( 𝜓 ↔ 𝜃 ) )
2		keephyp.2	⊢ ( 𝐵 = if ( 𝜑 , 𝐴 , 𝐵 ) → ( 𝜒 ↔ 𝜃 ) )
3		keephyp.3	⊢ 𝜓
4		keephyp.4	⊢ 𝜒
5	1 2	ifboth	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
6	3 4 5	mp2an	⊢ 𝜃