Theorem eqs2 829

Description: Lemma used in proofs of substitution properties.

Assertion

Ref	Expression
eqs2	|- (-. A.x x = y -> (-. A.x(x = y -> -. ph) -> A.x(x = y -> ph)))

Proof of Theorem eqs2

Step	Hyp	Ref	Expression
1		eq6 826	. 2 |- (-. A.x x = y -> A.x -. A.x x = y)
2		hbn1 708	. 2 |- (-. A.x(x = y -> -. ph) -> A.x -. A.x(x = y -> -. ph))
3		ax-11 801	. . . 4 |- (-. A.x x = y -> (x = y -> (-. ph -> A.x(x = y -> -. ph))))
4		con1 84	. . . 4 |- ((-. ph -> A.x(x = y -> -. ph)) -> (-. A.x(x = y -> -. ph) -> ph))
5	3, 4	syl6 23	. . 3 |- (-. A.x x = y -> (x = y -> (-. A.x(x = y -> -. ph) -> ph)))
6	5	com23 32	. 2 |- (-. A.x x = y -> (-. A.x(x = y -> -. ph) -> (x = y -> ph)))
7	1, 2, 6	19.21ad 741	1 |- (-. A.x x = y -> (-. A.x(x = y -> -. ph) -> A.x(x = y -> ph)))

Syntax hints:  -. wn 1   -> wi 2  A.wal 672   = weq 797

This theorem is referenced by:  eqs5 832

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

metamath.org