Theorem ax16b 2749

Description: This theorem shows that axiom ax-16 1210 is redundant in the presence of theorem dtruALT 2748, 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 2748	. 2 |- -. A.x x = y
2	1	pm2.21i 77	1 |- (A.x x = y -> (ph -> A.xph))

Syntax hints:   -> wi 3  A.wal 954   = wceq 956

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-nul 2710  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981

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