Theorem ax13b 1964

Description: An equivalence between two ways of expressing ax-13 2246. See the comment for ax-13 2246. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) (Revised by BJ, 15-Sep-2020.)

Assertion

Ref	Expression
ax13b	|- ( ( -. x = y -> ( y = z -> ph ) ) <-> ( -. x = y -> ( -. x = z -> ( y = z -> ph ) ) ) )

Proof of Theorem ax13b

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 |- ( ( y = z -> ph ) -> ( -. x = z -> ( y = z -> ph ) ) )
2		equeuclr 1950	. . . . . 6 |- ( y = z -> ( x = z -> x = y ) )
3	2	con3rr3 151	. . . . 5 |- ( -. x = y -> ( y = z -> -. x = z ) )
4	3	imim1d 82	. . . 4 |- ( -. x = y -> ( ( -. x = z -> ( y = z -> ph ) ) -> ( y = z -> ( y = z -> ph ) ) ) )
5		pm2.43 56	. . . 4 |- ( ( y = z -> ( y = z -> ph ) ) -> ( y = z -> ph ) )
6	4, 5	syl6 35	. . 3 |- ( -. x = y -> ( ( -. x = z -> ( y = z -> ph ) ) -> ( y = z -> ph ) ) )
7	1, 6	impbid2 216	. 2 |- ( -. x = y -> ( ( y = z -> ph ) <-> ( -. x = z -> ( y = z -> ph ) ) ) )
8	7	pm5.74i 260	1 |- ( ( -. x = y -> ( y = z -> ph ) ) <-> ( -. x = y -> ( -. x = z -> ( y = z -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196

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:  ax13  2249  ax13ALT  2305  ax13fromc9  34191