Theorem ax10w 2165

Description: Weak version of ax-10 2177 from which we can prove any ax-10 2177 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. It is an alias of hbn1w 2070 introduced for labeling consistency. (Contributed by NM, 9-Apr-2017.) Use hbn1w 2070 instead. (New usage is discouraged.)

Hypothesis

Ref	Expression
ax10w.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ax10w	⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

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

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem ax10w

Step	Hyp	Ref	Expression
1		ax10w.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	hbn1w 2070	1 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802

This theorem is referenced by: (None)