Theorem keephyp 4419

Description: Transform a hypothesis 𝜓 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.)

Hypotheses

Ref	Expression
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

Ref	Expression
keephyp	⊢ 𝜃

Proof of Theorem keephyp

Step	Hyp	Ref	Expression
1		keephyp.3	. 2 ⊢ 𝜓
2		keephyp.4	. 2 ⊢ 𝜒
3		keephyp.1	. . 3 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃))
4		keephyp.2	. . 3 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃))
5	3, 4	ifboth 4388	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
6	1, 2, 5	mp2an 679	1 ⊢ 𝜃

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1507  ifcif 4350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-ext 2750

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-if 4351

This theorem is referenced by:  boxcutc  8302  fin23lem13  9552  abvtrivd  19333  znf1o  20400  zntoslem  20405  dscmet  22885  sqff1o  25461  lgsne0  25613  dchrisum0flblem1  25786  dchrisum0flblem2  25787