Theorem gencbvex 2902

Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

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

Assertion

Ref	Expression
gencbvex	⊢ (∃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 gencbvex

Step	Hyp	Ref	Expression
1		excom 1741	. 2 ⊢ (∃x∃y(y = A ∧ (θ ∧ ψ)) ↔ ∃y∃x(y = A ∧ (θ ∧ ψ)))
2		gencbvex.1	. . . 4 ⊢ A ∈ V
3		gencbvex.3	. . . . . . 7 ⊢ (A = y → (χ ↔ θ))
4		gencbvex.2	. . . . . . 7 ⊢ (A = y → (φ ↔ ψ))
5	3, 4	anbi12d 691	. . . . . 6 ⊢ (A = y → ((χ ∧ φ) ↔ (θ ∧ ψ)))
6	5	bicomd 192	. . . . 5 ⊢ (A = y → ((θ ∧ ψ) ↔ (χ ∧ φ)))
7	6	eqcoms 2356	. . . 4 ⊢ (y = A → ((θ ∧ ψ) ↔ (χ ∧ φ)))
8	2, 7	ceqsexv 2895	. . 3 ⊢ (∃y(y = A ∧ (θ ∧ ψ)) ↔ (χ ∧ φ))
9	8	exbii 1582	. 2 ⊢ (∃x∃y(y = A ∧ (θ ∧ ψ)) ↔ ∃x(χ ∧ φ))
10		19.41v 1901	. . . 4 ⊢ (∃x(y = A ∧ (θ ∧ ψ)) ↔ (∃x y = A ∧ (θ ∧ ψ)))
11		simpr 447	. . . . 5 ⊢ ((∃x y = A ∧ (θ ∧ ψ)) → (θ ∧ ψ))
12		gencbvex.4	. . . . . . . 8 ⊢ (θ ↔ ∃x(χ ∧ A = y))
13		eqcom 2355	. . . . . . . . . . 11 ⊢ (A = y ↔ y = A)
14	13	biimpi 186	. . . . . . . . . 10 ⊢ (A = y → y = A)
15	14	adantl 452	. . . . . . . . 9 ⊢ ((χ ∧ A = y) → y = A)
16	15	eximi 1576	. . . . . . . 8 ⊢ (∃x(χ ∧ A = y) → ∃x y = A)
17	12, 16	sylbi 187	. . . . . . 7 ⊢ (θ → ∃x y = A)
18	17	adantr 451	. . . . . 6 ⊢ ((θ ∧ ψ) → ∃x y = A)
19	18	ancri 535	. . . . 5 ⊢ ((θ ∧ ψ) → (∃x y = A ∧ (θ ∧ ψ)))
20	11, 19	impbii 180	. . . 4 ⊢ ((∃x y = A ∧ (θ ∧ ψ)) ↔ (θ ∧ ψ))
21	10, 20	bitri 240	. . 3 ⊢ (∃x(y = A ∧ (θ ∧ ψ)) ↔ (θ ∧ ψ))
22	21	exbii 1582	. 2 ⊢ (∃y∃x(y = A ∧ (θ ∧ ψ)) ↔ ∃y(θ ∧ ψ))
23	1, 9, 22	3bitr3i 266	1 ⊢ (∃x(χ ∧ φ) ↔ ∃y(θ ∧ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  gencbvex2  2903  gencbval  2904