Theorem abbid 1576

Description: Equivalent wff's yield equal class abstractions (deduction rule).

Hypotheses

Ref	Expression
abbid.1	|- (ph -> A.xph)
abbid.2	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
abbid	|- (ph -> {x | ps} = {x | ch})

Proof of Theorem abbid

Step	Hyp	Ref	Expression
1		abbid.1	. . 3 |- (ph -> A.xph)
2		abbid.2	. . 3 |- (ph -> (ps <-> ch))
3	1, 2	19.21ai 998	. 2 |- (ph -> A.x(ps <-> ch))
4		eq2ab 1573	. 2 |- ({x | ps} = {x | ch} <-> A.x(ps <-> ch))
5	3, 4	sylibr 200	1 |- (ph -> {x | ps} = {x | ch})

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

This theorem is referenced by:  abbidv 1577  rabeqf 1808  abidhb 1912  opabbid 2669  moabex 2766

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

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