Theorem bnj118 30939

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj118.1	|- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj118.2	|- ( ph' <-> [. 1o / n ]. ph )

Assertion

Ref	Expression
bnj118	|- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )

Distinct variable groups:   A, n   R, n   f, n   x, n

Allowed substitution hints:   ph( x, f, n)   A( x, f)   R( x, f)   ph'( x, f, n)

Proof of Theorem bnj118

Step	Hyp	Ref	Expression
1		bnj118.2	. 2 |- ( ph' <-> [. 1o / n ]. ph )
2		bnj118.1	. . 3 |- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
3		bnj105 30790	. . 3 |- 1o e. _V
4	2, 3	bnj91 30931	. 2 |- ( [. 1o / n ]. ph <-> ( f ` (/) ) = pred ( x , A , R ) )
5	1, 4	bitri 264	1 |- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )

Syntax hints:   <-> wb 196   = wceq 1483   [.wsbc 3435   (/)c0 3915   ` cfv 5888   1oc1o 7553   predc-bnj14 30754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-suc 5729  df-1o 7560

This theorem is referenced by:  bnj151  30947  bnj153  30950