Theorem eqs2 829

Description: Lemma used in proofs of substitution properties.

Assertion

Ref	Expression
eqs2	⊢ (¬ ∀x x = y → (¬ ∀x(x = y → ¬ φ) → ∀x(x = y → φ)))

Proof of Theorem eqs2

Step	Hyp	Ref	Expression
1		eq6 826	. 2 ⊢ (¬ ∀x x = y → ∀x ¬ ∀x x = y)
2		hbn1 708	. 2 ⊢ (¬ ∀x(x = y → ¬ φ) → ∀x ¬ ∀x(x = y → ¬ φ))
3		ax-11 801	. . . 4 ⊢ (¬ ∀x x = y → (x = y → (¬ φ → ∀x(x = y → ¬ φ))))
4		con1 84	. . . 4 ⊢ ((¬ φ → ∀x(x = y → ¬ φ)) → (¬ ∀x(x = y → ¬ φ) → φ))
5	3, 4	syl6 23	. . 3 ⊢ (¬ ∀x x = y → (x = y → (¬ ∀x(x = y → ¬ φ) → φ)))
6	5	com23 32	. 2 ⊢ (¬ ∀x x = y → (¬ ∀x(x = y → ¬ φ) → (x = y → φ)))
7	1, 2, 6	19.21ad 741	1 ⊢ (¬ ∀x x = y → (¬ ∀x(x = y → ¬ φ) → ∀x(x = y → φ)))

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

This theorem is referenced by:  eqs5 832

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-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

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

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