Theorem cbviinvg 5043

Description: Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2377. Usage of the weaker cbviinv 5041 is preferred. (Contributed by Jeff Hankins, 26-Aug-2009.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
cbviinvg	⊢ ∩ 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 cbviinvg

Step	Hyp	Ref	Expression
1		nfcv 2905	. 2 ⊢ Ⅎ𝑦𝐵
2		nfcv 2905	. 2 ⊢ Ⅎ𝑥𝐶
3		cbviunvg.1	. 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
4	1, 2, 3	cbviing 5039	1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1540  ∩ ciin 4992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-iin 4994

This theorem is referenced by: (None)