Theorem dvelim 2016

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 normally has z free and can be read φ(z), and ψ substitutes y for z and can be read φ(y). We don't require that x and y be distinct: if they aren't, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with ∀x∀z, conjoin them, and apply dvelimdf 2082.

Other variants of this theorem are dvelimh 1964 (with no distinct variable restrictions), dvelimhw 1849 (that avoids ax-12 1925), and dvelimALT 2133 (that avoids ax-10 2140). (Contributed by NM, 23-Nov-1994.)

Hypotheses

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

Assertion

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

Distinct variable group:   ψ,z

Allowed substitution hints:   φ(x,y,z)   ψ(x,y)

Proof of Theorem dvelim

Step	Hyp	Ref	Expression
1		dvelim.1	. 2 ⊢ (φ → ∀xφ)
2		ax-17 1616	. 2 ⊢ (ψ → ∀zψ)
3		dvelim.2	. 2 ⊢ (z = y → (φ ↔ ψ))
4	1, 2, 3	dvelimh 1964	1 ⊢ (¬ ∀x x = y → (ψ → ∀xψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax15  2021  eujustALT  2207