Theorem spALT 40603

Description: sp 2182 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2182 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 2341	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑))
3		axc10 2403	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
4	2, 3	syl 17	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)