Theorem keephyp 4560

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 4528	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
6	1, 2, 5	mp2an 692	1 ⊢ 𝜃

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ifcif 4488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-if 4489

This theorem is referenced by:  boxcutc  8914  fin23lem13  10285  abvtrivd  20741  znf1o  21461  zntoslem  21466  dscmet  24460  sqff1o  27092  lgsne0  27246  dchrisum0flblem1  27419  dchrisum0flblem2  27420