Theorem keephyp 2367

Description: Transform a hypothesis ps that we want to keep (but contains the same class variable A used in the eliminated hypothesis) for use with the weak deduction theorem.

Hypotheses

Ref	Expression
keephyp.1	|- (A = if(ph, A, B) -> (ps <-> th))
keephyp.2	|- (B = if(ph, A, B) -> (ch <-> th))
keephyp.3	|- ps
keephyp.4	|- ch

Assertion

Ref	Expression
keephyp	|- th

Proof of Theorem keephyp

Step	Hyp	Ref	Expression
1		keephyp.3	. 2 |- ps
2		keephyp.4	. 2 |- ch
3		keephyp.1	. . 3 |- (A = if(ph, A, B) -> (ps <-> th))
4		keephyp.2	. . 3 |- (B = if(ph, A, B) -> (ch <-> th))
5	3, 4	ifboth 2346	. 2 |- ((ps /\ ch) -> th)
6	1, 2, 5	mp2an 694	1 |- th

Syntax hints:   -> wi 3   <-> wb 146   = wceq 1099  ifcif 2332

This theorem is referenced by:  keepel 2370  mulcant2 5611  sqrlem21 6574  sqrlem22 6575  projlem7 9322

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-if 2333

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