Theorem cbvexdh 1896

Description: Deduction used to change bound variables, using implicit substitition, particularly useful in conjunction with dvelim 1990. (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 1506	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		ax-17 1506	. . . 4 ⊢ (𝜒 → ∀𝑥𝜒)
3	2	hbex 1615	. . 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 1680	. . . . . . 7 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
8		bicom1 130	. . . . . . 7 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 ↔ 𝜓))
9	7, 8	imim12i 59	. . . . . 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 1707	. . . 4 ⊢ (𝜑 → (∃𝑦(𝑦 = 𝑥 ∧ 𝜒) ↔ 𝜓))
12		simpr 109	. . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝜒) → 𝜒)
13	12	eximi 1579	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝜒) → ∃𝑦𝜒)
14	11, 13	syl6bir 163	. . 3 ⊢ (𝜑 → (𝜓 → ∃𝑦𝜒))
15	1, 3, 14	exlimdh 1575	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑦𝜒))
16	1, 5	eximdh 1590	. . . 4 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥∀𝑦𝜓))
17		19.12 1643	. . . 4 ⊢ (∃𝑥∀𝑦𝜓 → ∀𝑦∃𝑥𝜓)
18	16, 17	syl6 33	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑦∃𝑥𝜓))
19	2	a1i 9	. . . . 5 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
20	1, 19, 6	equsexd 1707	. . . 4 ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒))
21		simpr 109	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜓) → 𝜓)
22	21	eximi 1579	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑥𝜓)
23	20, 22	syl6bir 163	. . 3 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓))
24	4, 18, 23	exlimd2 1574	. 2 ⊢ (𝜑 → (∃𝑦𝜒 → ∃𝑥𝜓))
25	15, 24	impbid 128	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  cbvexd  1897