Theorem ax12a2-o 39765

Description: Derive ax-c15 39704 from a hypothesis in the form of ax-12 2216, without using ax-12 2216 or ax-c15 39704. The hypothesis is weaker than ax-12 2216, 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 2216, if we also have ax-c11 39702, which this proof uses. As Theorem ax12 2458 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 39703 instead of ax-c11 39702. (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 1943	. . 3 ⊢ (𝜑 → ∀𝑧𝜑)
2		ax12a2-o.1	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
3	1, 2	syl5 35	. 2 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
4	3	ax12v2-o 39764	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

This proof depends on syntax axioms:  ¬ wn 3   → 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-11 2195  ax-12 2216  ax-13 2407  ax-c5 39698  ax-c4 39699  ax-c7 39700  ax-c10 39701  ax-c11 39702  ax-c9 39705

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)