Theorem keephyp 4546

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 4514	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
6	1, 2, 5	mp2an 700	1 ⊢ 𝜃

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1554  ifcif 4474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-if 4475

This theorem is referenced by:  boxcutc  8912  fin23lem13  10279  abvtrivd  20854  znf1o  21576  zntoslem  21581  dscmet  24605  sqff1o  27216  lgsne0  27369  dchrisum0flblem1  27542  dchrisum0flblem2  27543