Theorem axc16nfALT 2469

Description: Alternate proof of axc16nf 2301, shorter but requiring ax-11 2197 and ax-13 2404. (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 2465	. 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦
2		axc16g 2298	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
3	1, 2	nf5d 2321	1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817

This theorem is used by: (None)