Theorem ax16b 2717

Description: This theorem shows that axiom ax-16 1194 is redundant in the presence of theorem dtruALT 2716, which states simply that at least two things exist. This justifies the remark at http://us.metamath.org/mpegif/mmzfcnd.html#twoness (which links to this theorem).

Assertion

Ref	Expression
ax16b	|- (A.x x = y -> (ph -> A.xph))

Distinct variable group:   x,y

Proof of Theorem ax16b

Step	Hyp	Ref	Expression
1		dtruALT 2716	. 2 |- -. A.x x = y
2	1	pm2.21i 77	1 |- (A.x x = y -> (ph -> A.xph))

Syntax hints:   -> wi 3  A.wal 950   = wceq 1099

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-12 1104  ax-13 1107  ax-14 1108  ax-17 1190  ax-nul 2678  ax-pow 2710

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957

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