Theorem eqs3 830

Description: Lemma used in proofs of substitution properties.

Assertion

Ref	Expression
eqs3	⊢ (∃x(x = y ∧ φ) ↔ ¬ ∀x(x = y → ¬ φ))

Proof of Theorem eqs3

Step	Hyp	Ref	Expression
1		alinexa 724	. 2 ⊢ (∀x(x = y → ¬ φ) ↔ ¬ ∃x(x = y ∧ φ))
2	1	bicon2i 194	1 ⊢ (∃x(x = y ∧ φ) ↔ ¬ ∀x(x = y → ¬ φ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797

This theorem is referenced by:  eqs4 831  eqs5 832  sbn1 880  sbn2 881  sb5y 883

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-gen 677

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

metamath.org