Theorem sbeqalb 3488

Description: Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)

Assertion

Ref	Expression
sbeqalb	|- ( A e. V -> ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A = B ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem sbeqalb

Step	Hyp	Ref	Expression
1		bibi1 341	. . . . 5 |- ( ( ph <-> x = A ) -> ( ( ph <-> x = B ) <-> ( x = A <-> x = B ) ) )
2	1	biimpa 501	. . . 4 |- ( ( ( ph <-> x = A ) /\ ( ph <-> x = B ) ) -> ( x = A <-> x = B ) )
3	2	biimpd 219	. . 3 |- ( ( ( ph <-> x = A ) /\ ( ph <-> x = B ) ) -> ( x = A -> x = B ) )
4	3	alanimi 1744	. 2 |- ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A. x ( x = A -> x = B ) )
5		sbceqal 3487	. 2 |- ( A e. V -> ( A. x ( x = A -> x = B ) -> A = B ) )
6	4, 5	syl5 34	1 |- ( A e. V -> ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990

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

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:  iotaval  5862