Theorem eqs5 832

Description: Lemma used in proofs of substitution properties.

Assertion

Ref	Expression
eqs5	⊢ (¬ ∀x x = y → (∃x(x = y ∧ φ) → ∀x(x = y → φ)))

Proof of Theorem eqs5

Step	Hyp	Ref	Expression
1		eqs2 829	. 2 ⊢ (¬ ∀x x = y → (¬ ∀x(x = y → ¬ φ) → ∀x(x = y → φ)))
2		eqs3 830	. 2 ⊢ (∃x(x = y ∧ φ) ↔ ¬ ∀x(x = y → ¬ φ))
3	1, 2	syl5ib 181	1 ⊢ (¬ ∀x x = y → (∃x(x = y ∧ φ) → ∀x(x = y → φ)))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797

This theorem is referenced by:  sb3 860  sb4 861  alexeq 1409

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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