Theorem keephyp 4557

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 4525	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
6	1, 2, 5	mp2an 705	1 ⊢ 𝜃

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ifcif 4485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-if 4486

This theorem is used by:  boxcutc  8951  fin23lem13  10337  abvtrivd  21002  znf1o  21768  zntoslem  21773  dscmet  24802  sqff1o  27419  lgsne0  27572  dchrisum0flblem1  27745  dchrisum0flblem2  27746