Theorem axc16nfOLD 2163

Description: Obsolete proof of axc16nf 2137 as of 12-Oct-2021. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2034. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
axc16nfOLD	⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem axc16nfOLD

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aev 1983	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑤)
2		nfa1 2028	. . 3 ⊢ Ⅎ𝑧∀𝑧 𝑧 = 𝑤
3		axc16 2135	. . 3 ⊢ (∀𝑧 𝑧 = 𝑤 → (𝜑 → ∀𝑧𝜑))
4	2, 3	nf5d 2118	. 2 ⊢ (∀𝑧 𝑧 = 𝑤 → Ⅎ𝑧𝜑)
5	1, 4	syl 17	1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Syntax hints:   → wi 4  ∀wal 1481  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)