Theorem cbvrexdva2 3458

Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2023.)

Hypotheses

Ref	Expression
cbvraldva2.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
cbvraldva2.2	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvrexdva2	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))

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

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrexdva2

Step	Hyp	Ref	Expression
1		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
2		cbvraldva2.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
3	1, 2	eleq12d 2907	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4		cbvraldva2.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
5	3, 4	anbi12d 632	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐵 ∧ 𝜒)))
6	5	ancoms 461	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐵 ∧ 𝜒)))
7	6	pm5.32da 581	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜒))))
8	7	cbvexvw 2040	. . . 4 ⊢ (∃𝑥(𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ ∃𝑦(𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜒)))
9		19.42v 1950	. . . 4 ⊢ (∃𝑥(𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝜑 ∧ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)))
10		19.42v 1950	. . . 4 ⊢ (∃𝑦(𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜒)) ↔ (𝜑 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒)))
11	8, 9, 10	3bitr3i 303	. . 3 ⊢ ((𝜑 ∧ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝜑 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒)))
12		pm5.32 576	. . 3 ⊢ ((𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒))) ↔ ((𝜑 ∧ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝜑 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒))))
13	11, 12	mpbir 233	. 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒)))
14		df-rex 3144	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
15		df-rex 3144	. 2 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒))
16	13, 14, 15	3bitr4g 316	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∃wrex 3139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-cleq 2814  df-clel 2893  df-rex 3144

This theorem is referenced by:  cbvrexdva  3461  mreexexlemd  16909  eulerpartlemgvv  31629  ismnu  40590