Theorem dvelimdf 2082

Description: Deduction form of dvelimf 1997. This version may be useful if we want to avoid ax-17 1616 and use ax-16 2144 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
dvelimdf.1	⊢ Ⅎxφ
dvelimdf.2	⊢ Ⅎzφ
dvelimdf.3	⊢ (φ → Ⅎxψ)
dvelimdf.4	⊢ (φ → Ⅎzχ)
dvelimdf.5	⊢ (φ → (z = y → (ψ ↔ χ)))

Assertion

Ref	Expression
dvelimdf	⊢ (φ → (¬ ∀x x = y → Ⅎxχ))

Proof of Theorem dvelimdf

Step	Hyp	Ref	Expression
1		dvelimdf.2	. . . . . 6 ⊢ Ⅎzφ
2		dvelimdf.3	. . . . . 6 ⊢ (φ → Ⅎxψ)
3	1, 2	alrimi 1765	. . . . 5 ⊢ (φ → ∀zℲxψ)
4		nfsb4t 2080	. . . . 5 ⊢ (∀zℲxψ → (¬ ∀x x = y → Ⅎx[y / z]ψ))
5	3, 4	syl 15	. . . 4 ⊢ (φ → (¬ ∀x x = y → Ⅎx[y / z]ψ))
6	5	imp 418	. . 3 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx[y / z]ψ)
7		dvelimdf.1	. . . . 5 ⊢ Ⅎxφ
8		nfnae 1956	. . . . 5 ⊢ Ⅎx ¬ ∀x x = y
9	7, 8	nfan 1824	. . . 4 ⊢ Ⅎx(φ ∧ ¬ ∀x x = y)
10		dvelimdf.4	. . . . . 6 ⊢ (φ → Ⅎzχ)
11		dvelimdf.5	. . . . . 6 ⊢ (φ → (z = y → (ψ ↔ χ)))
12	1, 10, 11	sbied 2036	. . . . 5 ⊢ (φ → ([y / z]ψ ↔ χ))
13	12	adantr 451	. . . 4 ⊢ ((φ ∧ ¬ ∀x x = y) → ([y / z]ψ ↔ χ))
14	9, 13	nfbidf 1774	. . 3 ⊢ ((φ ∧ ¬ ∀x x = y) → (Ⅎx[y / z]ψ ↔ Ⅎxχ))
15	6, 14	mpbid 201	. 2 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxχ)
16	15	ex 423	1 ⊢ (φ → (¬ ∀x x = y → Ⅎxχ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544  [wsb 1648

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  df-sb 1649

This theorem is referenced by:  dvelimdc  2509