Theorem cbvixpv 6992

Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypothesis

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

Assertion

Ref	Expression
cbvixpv	⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

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

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

Proof of Theorem cbvixpv

Step	Hyp	Ref	Expression
1		nfcv 2392	. 2 ⊢ Ⅎ𝑦𝐵
2		nfcv 2392	. 2 ⊢ Ⅎ𝑥𝐶
3		cbvixpv.1	. 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
4	1, 2, 3	cbvixp 6991	1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1402  Xcixp 6974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-iota 5335  df-fn 5378  df-fv 5383  df-ixp 6975

This theorem is referenced by:  depindlem2  16731