Theorem ax16b 2739

Description: This theorem shows that axiom ax-16 1206 is redundant in the presence of theorem dtruALT 2738, 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 2738	. 2 |- -. A.x x = y
2	1	pm2.21i 77	1 |- (A.x x = y -> (ph -> A.xph))

Syntax hints:   -> wi 3  A.wal 951   = wceq 953

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-nul 2700  ax-pow 2732

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978

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