Theorem ax10fromc7 34180

Description: Rederivation of axiom ax-10 2019 from ax-c7 34170, ax-c4 34169, ax-c5 34168, ax-gen 1722 and propositional calculus. See axc7 2132 for the derivation of ax-c7 34170 from ax-10 2019. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax10fromc7	⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Proof of Theorem ax10fromc7

Step	Hyp	Ref	Expression
1		ax-c4 34169	. . 3 ⊢ (∀𝑥(∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑))
2		ax-c5 34168	. . . 4 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥∀𝑥𝜑)
3		ax-c4 34169	. . . . 5 ⊢ (∀𝑥(∀𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑))
4		id 22	. . . . 5 ⊢ (∀𝑥𝜑 → ∀𝑥𝜑)
5	3, 4	mpg 1724	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
6	2, 5	nsyl 135	. . 3 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥𝜑)
7	1, 6	mpg 1724	. 2 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
8		ax-c7 34170	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥𝜑)
9	7, 8	nsyl4 156	1 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-c5 34168  ax-c4 34169  ax-c7 34170

This theorem is referenced by:  hba1-o  34182  axc5c711  34203  equidq  34209