Theorem dedhb 2426

Description: A deduction theorem for converting the inference |- (y e. A -> A.xy e. A) => |- ph into a closed theorem. Use hba1 1350 and hbab 1875 to eliminate the hypothesis of the substitution instance ps of the inference. For converting the inference form into a deduction form, abidhb 2423 is useful.

Hypotheses

Ref	Expression
dedhb.1	|- (A = {z | A.x z e. A} -> (ph <-> ps))
dedhb.2	|- ps

Assertion

Ref	Expression
dedhb	|- (A.y(y e. A -> A.x y e. A) -> ph)

Distinct variable groups:   y,A   x,z   x,y   z,A

Proof of Theorem dedhb

Step	Hyp	Ref	Expression
1		dedhb.2	. 2 |- ps
2		abidhb 2423	. . . 4 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
3	2	eqcomd 1889	. . 3 |- (A.y(y e. A -> A.x y e. A) -> A = {z | A.x z e. A})
4		dedhb.1	. . 3 |- (A = {z | A.x z e. A} -> (ph <-> ps))
5	3, 4	syl 12	. 2 |- (A.y(y e. A -> A.x y e. A) -> (ph <-> ps))
6	1, 5	mpbiri 211	1 |- (A.y(y e. A -> A.x y e. A) -> ph)

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  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

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