Theorem ax12a2-o 37268

Description: Derive ax-c15 37207 from a hypothesis in the form of ax-12 2171, without using ax-12 2171 or ax-c15 37207. The hypothesis is weaker than ax-12 2171, with 𝑧 both distinct from 𝑥 and not occurring in 𝜑. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-12 2171, if we also have ax-c11 37205, which this proof uses. As Theorem ax12 2422 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 37206 instead of ax-c11 37205. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax12a2-o.1	⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))

Assertion

Ref	Expression
ax12a2-o	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax12a2-o

Step	Hyp	Ref	Expression
1		ax-5 1913	. . 3 ⊢ (𝜑 → ∀𝑧𝜑)
2		ax12a2-o.1	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
3	1, 2	syl5 34	. 2 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
4	3	ax12v2-o 37267	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371  ax-c5 37201  ax-c4 37202  ax-c7 37203  ax-c10 37204  ax-c11 37205  ax-c9 37208

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)