Theorem dedhb 1915

Description: A deduction theorem for converting the inference |- (y e. A -> A.xy e. A) => |- ph into a closed theorem. Use hba1 1003 and hbab 1467 to eliminate the hypothesis of the substitution instance ps of the inference.

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,z,A   x,y,z

Proof of Theorem dedhb

Step	Hyp	Ref	Expression
1		dedhb.2	. 2 |- ps
2		abidhb 1912	. . . 4 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
3	2	eqcomd 1480	. . 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 10	. 2 |- (A.y(y e. A -> A.x y e. A) -> (ph <-> ps))
6	1, 5	mpbiri 194	1 |- (A.y(y e. A -> A.x y e. A) -> ph)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  {cab 1463

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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