Theorem ax16b 2745

Description: This theorem shows that axiom ax-16 1209 is redundant in the presence of theorem dtruALT 2744, 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	⊢ (∀x x = y → (φ → ∀xφ))

Distinct variable group:   x,y

Proof of Theorem ax16b

Step	Hyp	Ref	Expression
1		dtruALT 2744	. 2 ⊢ ¬ ∀x x = y
2	1	pm2.21i 77	1 ⊢ (∀x x = y → (φ → ∀xφ))

Syntax hints:   → wi 3  ∀wal 953   = wceq 955

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-nul 2706  ax-pow 2738

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980

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