Theorem gencbval 2778

Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.)

Hypotheses

Ref	Expression
gencbval.1	⊢ 𝐴 ∈ V
gencbval.2	⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓))
gencbval.3	⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃))
gencbval.4	⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))

Assertion

Ref	Expression
gencbval	⊢ (∀𝑥(𝜒 → 𝜑) ↔ ∀𝑦(𝜃 → 𝜓))

Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝜃,𝑥   𝜒,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥)   𝜃(𝑦)   𝐴(𝑥)

Proof of Theorem gencbval

Step	Hyp	Ref	Expression
1		alcom 1471	. 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝐴 → (𝜃 → 𝜓)) ↔ ∀𝑦∀𝑥(𝑦 = 𝐴 → (𝜃 → 𝜓)))
2		gencbval.1	. . . 4 ⊢ 𝐴 ∈ V
3		gencbval.3	. . . . . . 7 ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃))
4		gencbval.2	. . . . . . 7 ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	imbi12d 233	. . . . . 6 ⊢ (𝐴 = 𝑦 → ((𝜒 → 𝜑) ↔ (𝜃 → 𝜓)))
6	5	bicomd 140	. . . . 5 ⊢ (𝐴 = 𝑦 → ((𝜃 → 𝜓) ↔ (𝜒 → 𝜑)))
7	6	eqcoms 2173	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝜃 → 𝜓) ↔ (𝜒 → 𝜑)))
8	2, 7	ceqsalv 2760	. . 3 ⊢ (∀𝑦(𝑦 = 𝐴 → (𝜃 → 𝜓)) ↔ (𝜒 → 𝜑))
9	8	albii 1463	. 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝐴 → (𝜃 → 𝜓)) ↔ ∀𝑥(𝜒 → 𝜑))
10		19.23v 1876	. . . 4 ⊢ (∀𝑥(𝑦 = 𝐴 → (𝜃 → 𝜓)) ↔ (∃𝑥 𝑦 = 𝐴 → (𝜃 → 𝜓)))
11		gencbval.4	. . . . . . 7 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))
12		eqcom 2172	. . . . . . . . . 10 ⊢ (𝐴 = 𝑦 ↔ 𝑦 = 𝐴)
13	12	biimpi 119	. . . . . . . . 9 ⊢ (𝐴 = 𝑦 → 𝑦 = 𝐴)
14	13	adantl 275	. . . . . . . 8 ⊢ ((𝜒 ∧ 𝐴 = 𝑦) → 𝑦 = 𝐴)
15	14	eximi 1593	. . . . . . 7 ⊢ (∃𝑥(𝜒 ∧ 𝐴 = 𝑦) → ∃𝑥 𝑦 = 𝐴)
16	11, 15	sylbi 120	. . . . . 6 ⊢ (𝜃 → ∃𝑥 𝑦 = 𝐴)
17		pm2.04 82	. . . . . 6 ⊢ ((∃𝑥 𝑦 = 𝐴 → (𝜃 → 𝜓)) → (𝜃 → (∃𝑥 𝑦 = 𝐴 → 𝜓)))
18	16, 17	mpdi 43	. . . . 5 ⊢ ((∃𝑥 𝑦 = 𝐴 → (𝜃 → 𝜓)) → (𝜃 → 𝜓))
19		ax-1 6	. . . . 5 ⊢ ((𝜃 → 𝜓) → (∃𝑥 𝑦 = 𝐴 → (𝜃 → 𝜓)))
20	18, 19	impbii 125	. . . 4 ⊢ ((∃𝑥 𝑦 = 𝐴 → (𝜃 → 𝜓)) ↔ (𝜃 → 𝜓))
21	10, 20	bitri 183	. . 3 ⊢ (∀𝑥(𝑦 = 𝐴 → (𝜃 → 𝜓)) ↔ (𝜃 → 𝜓))
22	21	albii 1463	. 2 ⊢ (∀𝑦∀𝑥(𝑦 = 𝐴 → (𝜃 → 𝜓)) ↔ ∀𝑦(𝜃 → 𝜓))
23	1, 9, 22	3bitr3i 209	1 ⊢ (∀𝑥(𝜒 → 𝜑) ↔ ∀𝑦(𝜃 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by: (None)