Theorem cbvexdh 1798

Description: Deduction used to change bound variables, using implicit substitition, particularly useful in conjunction with dvelim 1890. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 30-Dec-2017.)

Hypotheses

Ref	Expression
cbvexdh.1	⊢ (φ → ∀yφ)
cbvexdh.2	⊢ (φ → (ψ → ∀yψ))
cbvexdh.3	⊢ (φ → (x = y → (ψ ↔ χ)))

Assertion

Ref	Expression
cbvexdh	⊢ (φ → (∃xψ ↔ ∃yχ))

Distinct variable groups:   φ,x   χ,x

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

Proof of Theorem cbvexdh

Step	Hyp	Ref	Expression
1		ax-17 1416	. . 3 ⊢ (φ → ∀xφ)
2		ax-17 1416	. . . 4 ⊢ (χ → ∀xχ)
3	2	hbex 1524	. . 3 ⊢ (∃yχ → ∀x∃yχ)
4		cbvexdh.1	. . . . 5 ⊢ (φ → ∀yφ)
5		cbvexdh.2	. . . . 5 ⊢ (φ → (ψ → ∀yψ))
6		cbvexdh.3	. . . . . 6 ⊢ (φ → (x = y → (ψ ↔ χ)))
7		equcomi 1589	. . . . . . 7 ⊢ (y = x → x = y)
8		bicom1 122	. . . . . . 7 ⊢ ((ψ ↔ χ) → (χ ↔ ψ))
9	7, 8	imim12i 53	. . . . . 6 ⊢ ((x = y → (ψ ↔ χ)) → (y = x → (χ ↔ ψ)))
10	6, 9	syl 14	. . . . 5 ⊢ (φ → (y = x → (χ ↔ ψ)))
11	4, 5, 10	equsexd 1614	. . . 4 ⊢ (φ → (∃y(y = x ∧ χ) ↔ ψ))
12		simpr 103	. . . . 5 ⊢ ((y = x ∧ χ) → χ)
13	12	eximi 1488	. . . 4 ⊢ (∃y(y = x ∧ χ) → ∃yχ)
14	11, 13	syl6bir 153	. . 3 ⊢ (φ → (ψ → ∃yχ))
15	1, 3, 14	exlimdh 1484	. 2 ⊢ (φ → (∃xψ → ∃yχ))
16	1, 5	eximdh 1499	. . . 4 ⊢ (φ → (∃xψ → ∃x∀yψ))
17		19.12 1552	. . . 4 ⊢ (∃x∀yψ → ∀y∃xψ)
18	16, 17	syl6 29	. . 3 ⊢ (φ → (∃xψ → ∀y∃xψ))
19	2	a1i 9	. . . . 5 ⊢ (φ → (χ → ∀xχ))
20	1, 19, 6	equsexd 1614	. . . 4 ⊢ (φ → (∃x(x = y ∧ ψ) ↔ χ))
21		simpr 103	. . . . 5 ⊢ ((x = y ∧ ψ) → ψ)
22	21	eximi 1488	. . . 4 ⊢ (∃x(x = y ∧ ψ) → ∃xψ)
23	20, 22	syl6bir 153	. . 3 ⊢ (φ → (χ → ∃xψ))
24	4, 18, 23	exlimd2 1483	. 2 ⊢ (φ → (∃yχ → ∃xψ))
25	15, 24	impbid 120	1 ⊢ (φ → (∃xψ ↔ ∃yχ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  cbvexd  1799