Theorem pm13.195 38614

Description: Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3460. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)

Assertion

Ref	Expression
pm13.195	|- ( E. y ( y = A /\ ph ) <-> [. A / y ]. ph )

Distinct variable group:   y, A

Allowed substitution hint:   ph( y)

Proof of Theorem pm13.195

Step	Hyp	Ref	Expression
1		sbc5 3460	. 2 |- ( [. A / y ]. ph <-> E. y ( y = A /\ ph ) )
2	1	bicomi 214	1 |- ( E. y ( y = A /\ ph ) <-> [. A / y ]. ph )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   [.wsbc 3435

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-12 2047  ax-13 2246  ax-ext 2602

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-v 3202  df-sbc 3436

This theorem is referenced by: (None)