Theorem keephyp 4527

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 4495	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
6	1, 2, 5	mp2an 688	1 ⊢ 𝜃

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  ifcif 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-if 4457

This theorem is referenced by:  boxcutc  8687  fin23lem13  10019  abvtrivd  20015  znf1o  20671  zntoslem  20676  dscmet  23634  sqff1o  26236  lgsne0  26388  dchrisum0flblem1  26561  dchrisum0flblem2  26562