Theorem ax10fromc7 37570

Description: Rederivation of Axiom ax-10 2137 from ax-c7 37560, ax-c4 37559, ax-c5 37558, ax-gen 1797 and propositional calculus. See axc7 2310 for the derivation of ax-c7 37560 from ax-10 2137. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-10 2137 instead. (New usage is discouraged.)

Assertion

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

Proof of Theorem ax10fromc7

Step	Hyp	Ref	Expression
1		ax-c4 37559	. . 3 ⊢ (∀𝑥(∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑))
2		ax-c5 37558	. . . 4 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥∀𝑥𝜑)
3		ax-c4 37559	. . . . 5 ⊢ (∀𝑥(∀𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑))
4		id 22	. . . . 5 ⊢ (∀𝑥𝜑 → ∀𝑥𝜑)
5	3, 4	mpg 1799	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
6	2, 5	nsyl 140	. . 3 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥𝜑)
7	1, 6	mpg 1799	. 2 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
8		ax-c7 37560	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥𝜑)
9	7, 8	nsyl4 158	1 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

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-c5 37558  ax-c4 37559  ax-c7 37560

This theorem is referenced by:  hba1-o  37572  axc5c711  37593  equidq  37599