Theorem cbviun 4004

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)

Hypotheses

Ref	Expression
cbviun.1	⊢ ℲyB
cbviun.2	⊢ ℲxC
cbviun.3	⊢ (x = y → B = C)

Assertion

Ref	Expression
cbviun	⊢ ∪x ∈ A B = ∪y ∈ A C

Distinct variable groups:   y,A   x,A

Allowed substitution hints:   B(x,y)   C(x,y)

Proof of Theorem cbviun

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviun.1	. . . . 5 ⊢ ℲyB
2	1	nfcri 2484	. . . 4 ⊢ Ⅎy z ∈ B
3		cbviun.2	. . . . 5 ⊢ ℲxC
4	3	nfcri 2484	. . . 4 ⊢ Ⅎx z ∈ C
5		cbviun.3	. . . . 5 ⊢ (x = y → B = C)
6	5	eleq2d 2420	. . . 4 ⊢ (x = y → (z ∈ B ↔ z ∈ C))
7	2, 4, 6	cbvrex 2833	. . 3 ⊢ (∃x ∈ A z ∈ B ↔ ∃y ∈ A z ∈ C)
8	7	abbii 2466	. 2 ⊢ {z ∣ ∃x ∈ A z ∈ B} = {z ∣ ∃y ∈ A z ∈ C}
9		df-iun 3972	. 2 ⊢ ∪x ∈ A B = {z ∣ ∃x ∈ A z ∈ B}
10		df-iun 3972	. 2 ⊢ ∪y ∈ A C = {z ∣ ∃y ∈ A z ∈ C}
11	8, 9, 10	3eqtr4i 2383	1 ⊢ ∪x ∈ A B = ∪y ∈ A C

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  ∃wrex 2616  ∪ciun 3970

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-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-iun 3972

This theorem is referenced by:  cbviunv  4006  funiunfvf  5469  fmpt2x  5731