Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ixpeq2d | Structured version Visualization version GIF version |
Description: Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
ixpeq2d.1 | ⊢ Ⅎ𝑥𝜑 |
ixpeq2d.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
ixpeq2d | ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpeq2d.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ixpeq2d.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
3 | 2 | ex 416 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)) |
4 | 1, 3 | ralrimi 3129 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) |
5 | ixpeq2 8534 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | |
6 | 4, 5 | syl 17 | 1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 Ⅎwnf 1790 ∈ wcel 2114 ∀wral 3054 Xcixp 8520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-12 2179 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-v 3402 df-in 3860 df-ss 3870 df-ixp 8521 |
This theorem is referenced by: hoicvrrex 43677 ovnlecvr 43679 ovnhoilem1 43722 hoi2toco 43728 ovnlecvr2 43731 opnvonmbllem1 43753 |
Copyright terms: Public domain | W3C validator |