Theorem dedth2v 2382

Description: Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 2385 is simpler to use. See also comments in dedth 2381.

Hypotheses

Ref	Expression
dedth2v.1	|- (A = if(ph, A, C) -> (ps <-> ch))
dedth2v.2	|- (B = if(ph, B, D) -> (ch <-> th))
dedth2v.3	|- th

Assertion

Ref	Expression
dedth2v	|- (ph -> ps)

Proof of Theorem dedth2v

Step	Hyp	Ref	Expression
1		dedth2v.3	. 2 |- th
2		iftrue 2364	. . . . 5 |- (ph -> if(ph, A, C) = A)
3	2	eqcomd 1479	. . . 4 |- (ph -> A = if(ph, A, C))
4		dedth2v.1	. . . 4 |- (A = if(ph, A, C) -> (ps <-> ch))
5	3, 4	syl 10	. . 3 |- (ph -> (ps <-> ch))
6		iftrue 2364	. . . . 5 |- (ph -> if(ph, B, D) = B)
7	6	eqcomd 1479	. . . 4 |- (ph -> B = if(ph, B, D))
8		dedth2v.2	. . . 4 |- (B = if(ph, B, D) -> (ch <-> th))
9	7, 8	syl 10	. . 3 |- (ph -> (ch <-> th))
10	5, 9	bitrd 527	. 2 |- (ph -> (ps <-> th))
11	1, 10	mpbiri 194	1 |- (ph -> ps)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955  ifcif 2359

This theorem is referenced by:  bcpasc2t 6936  climuni 7067  hlimuni 9097  omls 9234  osumlem8 9575  pjfot 9642  ghomgrplem 10380

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-if 2360

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