Theorem dedth2vOLD 3012

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 3015 is simpler to use. See also comments in dedth 3011.

Hypotheses

Ref	Expression
dedth2vOLD.1	|- (A = if(ph, A, C) -> (ps <-> ch))
dedth2vOLD.2	|- (B = if(ph, B, D) -> (ch <-> th))
dedth2vOLD.3	|- th

Assertion

Ref	Expression
dedth2vOLD	|- (ph -> ps)

Proof of Theorem dedth2vOLD

Step	Hyp	Ref	Expression
1		dedth2vOLD.3	. 2 |- th
2		iftrue 2989	. . . . 5 |- (ph -> if(ph, A, C) = A)
3	2	eqcomd 1889	. . . 4 |- (ph -> A = if(ph, A, C))
4		dedth2vOLD.1	. . . 4 |- (A = if(ph, A, C) -> (ps <-> ch))
5	3, 4	syl 12	. . 3 |- (ph -> (ps <-> ch))
6		iftrue 2989	. . . . 5 |- (ph -> if(ph, B, D) = B)
7	6	eqcomd 1889	. . . 4 |- (ph -> B = if(ph, B, D))
8		dedth2vOLD.2	. . . 4 |- (B = if(ph, B, D) -> (ch <-> th))
9	7, 8	syl 12	. . 3 |- (ph -> (ch <-> th))
10	5, 9	bitrd 587	. 2 |- (ph -> (ps <-> th))
11	1, 10	mpbiri 211	1 |- (ph -> ps)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298  ifcif 2982

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-if 2983

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