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Mirrors > Home > ILE Home > Th. List > ixpeq2dv | GIF version |
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
Ref | Expression |
---|---|
ixpeq2dv.1 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
ixpeq2dv | ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpeq2dv.1 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
2 | 1 | adantr 276 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
3 | 2 | ixpeq2dva 6731 | 1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 Xcixp 6716 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-in 3150 df-ss 3157 df-ixp 6717 |
This theorem is referenced by: prdsex 12740 |
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