Theorem spALT 44960

Description: sp 2222 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2222 instead. (New usage is discouraged.)

Assertion

Ref	Expression
spALT	⊢ (∀𝑥𝜑 → 𝜑)

Proof of Theorem spALT

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ (∀𝑥𝜑 → (𝑥 = 𝑦 → ∀𝑥𝜑))
2	1	axc4i 2357	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑))
3		axc10 2419	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
4	2, 3	syl 18	1 ⊢ (∀𝑥𝜑 → 𝜑)

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

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 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216  ax-13 2406

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

This theorem is used by: (None)