Theorem ax9 807

Description: This is a variant of ax-9 799. Axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint).

Assertion

Ref	Expression
ax9	⊢ (∀x(x = y → ∀xφ) → φ)

Proof of Theorem ax9

Step	Hyp	Ref	Expression
1		ax-9 799	. . . 4 ⊢ ¬ ∀x ¬ x = y
2		df-ex 679	. . . 4 ⊢ (∃x x = y ↔ ¬ ∀x ¬ x = y)
3	1, 2	mpbir 165	. . 3 ⊢ ∃x x = y
4		19.22 722	. . 3 ⊢ (∀x(x = y → ∀xφ) → (∃x x = y → ∃x∀xφ))
5	3, 4	mpi 44	. 2 ⊢ (∀x(x = y → ∀xφ) → ∃x∀xφ)
6		a6e 688	. 2 ⊢ (∃x∀xφ → φ)
7	5, 6	syl 12	1 ⊢ (∀x(x = y → ∀xφ) → φ)

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672  ∃wex 678   = weq 797

This theorem is referenced by:  ax9a 808  eqid 810  eqs1 828  eqsal 833  a4a 842  cbv1 845

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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