Theorem ddelim 1000

Description: This theorem can be used to eliminate a distinct variable restriction on x and z and replace it with the "distinctor" ¬ ∀xx = y as an antecedent. φ is considered to contain z, and ψ to have z replaced with y, and we don't require that x and y be distinct (if they aren't, the distinctor will become false and "protect" the consequent).

Hypotheses

Ref	Expression
ddelim.1	⊢ (φ → ∀xφ)
ddelim.2	⊢ (z = y → (φ ↔ ψ))

Assertion

Ref	Expression
ddelim	⊢ (¬ ∀x x = y → (ψ → ∀xψ))

Distinct variable group(s):   ψ,z

Proof of Theorem ddelim

Step	Hyp	Ref	Expression
1		ddelim.1	. 2 ⊢ (φ → ∀xφ)
2		ax-17 925	. 2 ⊢ (ψ → ∀zψ)
3		ddelim.2	. 2 ⊢ (z = y → (φ ↔ ψ))
4	1, 2, 3	ddelimf 908	1 ⊢ (¬ ∀x x = y → (ψ → ∀xψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∀wal 672   = weq 797

This theorem is referenced by:  ddeeq1 1001  ddeeq2 1002  ddeel1 1003  ddeel2 1004  ax15 1006  rgen2 1248  ralcom2 1314

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  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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