Theorem ax12a2-o 36090

Description: Derive ax-c15 36029 from a hypothesis in the form of ax-12 2176, without using ax-12 2176 or ax-c15 36029. The hypothesis is weaker than ax-12 2176, 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 2176, if we also have ax-c11 36027, which this proof uses. As theorem ax12 2444 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 36028 instead of ax-c11 36027. (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 1910	. . 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 36089	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176  ax-13 2389  ax-c5 36023  ax-c4 36024  ax-c7 36025  ax-c10 36026  ax-c11 36027  ax-c9 36030

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)