Theorem dvelim 1934

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 1933.

Other variants of this theorem are dvelimf 1932 (with no distinct variable restrictions) and dvelimALT 1927 (that avoids ax-10 1436). (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 1459	. 2 ⊢ (𝜓 → ∀𝑧𝜓)
3		dvelim.2	. 2 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	dvelimf 1932	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103  ∀wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by: (None)