Theorem bj-ssblem1 32630

Description: A lemma for the definiens of df-sb 1881. An instance of sp 2053 proved without it. Note: it has a common subproof with bj-ssbjust 32618. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssblem1	|- ( A. y ( y = t -> A. x ( x = y -> ph ) ) -> ( y = t -> A. x ( x = y -> ph ) ) )

Distinct variable groups:   x, y   y, t   ph, y

Allowed substitution hints:   ph( x, t)

Proof of Theorem bj-ssblem1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ1 1952	. . 3 |- ( y = z -> ( y = t <-> z = t ) )
2		equequ2 1953	. . . . 5 |- ( y = z -> ( x = y <-> x = z ) )
3	2	imbi1d 331	. . . 4 |- ( y = z -> ( ( x = y -> ph ) <-> ( x = z -> ph ) ) )
4	3	albidv 1849	. . 3 |- ( y = z -> ( A. x ( x = y -> ph ) <-> A. x ( x = z -> ph ) ) )
5	1, 4	imbi12d 334	. 2 |- ( y = z -> ( ( y = t -> A. x ( x = y -> ph ) ) <-> ( z = t -> A. x ( x = z -> ph ) ) ) )
6	5	spw 1967	1 |- ( A. y ( y = t -> A. x ( x = y -> ph ) ) -> ( y = t -> A. x ( x = y -> ph ) ) )

Syntax hints:   -> wi 4   A.wal 1481

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

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)