Theorem sb4 861

Description: One direction of a simplified definition of substitution when variables are distinct.

Assertion

Ref	Expression
sb4	⊢ (¬ ∀x x = y → ([y / x]φ → ∀x(x = y → φ)))

Proof of Theorem sb4

Step	Hyp	Ref	Expression
1		eqs5 832	. 2 ⊢ (¬ ∀x x = y → (∃x(x = y ∧ φ) → ∀x(x = y → φ)))
2		sb1 858	. 2 ⊢ ([y / x]φ → ∃x(x = y ∧ φ))
3	1, 2	syl5 22	1 ⊢ (¬ ∀x x = y → ([y / x]φ → ∀x(x = y → φ)))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  [wsb 852

This theorem is referenced by:  sb4b 862  sb6y 872  hbsb2 873  sbn1 880  sbi1 884  hbsb4 905  sbal1 996

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  df-sb 853

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