Theorem dvelim 2070

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 2069.

Other variants of this theorem are dvelimf 2068 (with no distinct variable restrictions) and dvelimALT 2063 (that avoids ax-10 1553). (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 1574	. 2 ⊢ (𝜓 → ∀𝑧𝜓)
3		dvelim.2	. 2 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	dvelimf 2068	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  ∀wal 1395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by: (None)