Theorem cbvrexdva2 2512

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.)

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 103	. . . . 5 ⊢ ((φ ∧ x = y) → x = y)
2		cbvraldva2.2	. . . . 5 ⊢ ((φ ∧ x = y) → A = B)
3	1, 2	eleq12d 2086	. . . 4 ⊢ ((φ ∧ x = y) → (x ∈ A ↔ y ∈ B))
4		cbvraldva2.1	. . . 4 ⊢ ((φ ∧ x = y) → (ψ ↔ χ))
5	3, 4	anbi12d 445	. . 3 ⊢ ((φ ∧ x = y) → ((x ∈ A ∧ ψ) ↔ (y ∈ B ∧ χ)))
6	5	cbvexdva 1782	. 2 ⊢ (φ → (∃x(x ∈ A ∧ ψ) ↔ ∃y(y ∈ B ∧ χ)))
7		df-rex 2286	. 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ))
8		df-rex 2286	. 2 ⊢ (∃y ∈ B χ ↔ ∃y(y ∈ B ∧ χ))
9	6, 7, 8	3bitr4g 212	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃y ∈ B χ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1226  ∃wex 1358   ∈ wcel 1370  ∃wrex 2281

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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-ext 2000

This theorem depends on definitions:  df-bi 110  df-nf 1326  df-cleq 2011  df-clel 2014  df-rex 2286

This theorem is referenced by:  cbvrexdva  2514  acexmid  5431