Theorem eqs1 828

Description: Lemma used in proofs of substitution properties.

Assertion

Ref	Expression
eqs1	⊢ (∀x(x = y → φ) → ¬ ∀x(x = y → ¬ φ))

Proof of Theorem eqs1

Step	Hyp	Ref	Expression
1		ax-4 673	. . . . 5 ⊢ (∀x(x = y → φ) → (x = y → φ))
2		ax-4 673	. . . . 5 ⊢ (∀x(x = y → ¬ φ) → (x = y → ¬ φ))
3	1, 2	msca 508	. . . 4 ⊢ (∀x(x = y → φ) → (x = y → ¬ ∀x(x = y → ¬ φ)))
4		hbn1 708	. . . 4 ⊢ (¬ ∀x(x = y → ¬ φ) → ∀x ¬ ∀x(x = y → ¬ φ))
5	3, 4	syl6 23	. . 3 ⊢ (∀x(x = y → φ) → (x = y → ∀x ¬ ∀x(x = y → ¬ φ)))
6	5	a5i 687	. 2 ⊢ (∀x(x = y → φ) → ∀x(x = y → ∀x ¬ ∀x(x = y → ¬ φ)))
7		ax9 807	. 2 ⊢ (∀x(x = y → ∀x ¬ ∀x(x = y → ¬ φ)) → ¬ ∀x(x = y → ¬ φ))
8	6, 7	syl 12	1 ⊢ (∀x(x = y → φ) → ¬ ∀x(x = y → ¬ φ))

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

This theorem is referenced by:  eqs4 831

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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