Theorem cbviunv 4006

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)

Hypothesis

Ref	Expression
cbviunv.1	⊢ (x = y → B = C)

Assertion

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

Distinct variable groups:   x,A   y,A   y,B   x,C

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

Proof of Theorem cbviunv

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲyB
2		nfcv 2490	. 2 ⊢ ℲxC
3		cbviunv.1	. 2 ⊢ (x = y → B = C)
4	1, 2, 3	cbviun 4004	1 ⊢ ∪x ∈ A B = ∪y ∈ A C

Syntax hints:   → wi 4   = wceq 1642  ∪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:  iunxdif2  4015