Theorem axc16nfALT 2453

Description: Alternate proof of axc16nf 2257, shorter but requiring ax-11 2154 and ax-13 2384. (Contributed by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc16nfALT	⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem axc16nfALT

Step	Hyp	Ref	Expression
1		nfae 2449	. 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦
2		axc16g 2254	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
3	1, 2	nf5d 2286	1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Syntax hints:   → wi 4  ∀wal 1529  Ⅎwnf 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-10 2139  ax-11 2154  ax-12 2170  ax-13 2384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779

This theorem is referenced by: (None)