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Mirrors > Home > MPE Home > Th. List > Mathboxes > altxpeq1 | Structured version Visualization version GIF version |
Description: Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.) |
Ref | Expression |
---|---|
altxpeq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ×× 𝐶) = (𝐵 ×× 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexeq 3319 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝑧 = ⟪𝑥, 𝑦⟫)) | |
2 | 1 | abbidv 2797 | . 2 ⊢ (𝐴 = 𝐵 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑧 = ⟪𝑥, 𝑦⟫} = {𝑧 ∣ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝑧 = ⟪𝑥, 𝑦⟫}) |
3 | df-altxp 35588 | . 2 ⊢ (𝐴 ×× 𝐶) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑧 = ⟪𝑥, 𝑦⟫} | |
4 | df-altxp 35588 | . 2 ⊢ (𝐵 ×× 𝐶) = {𝑧 ∣ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝑧 = ⟪𝑥, 𝑦⟫} | |
5 | 2, 3, 4 | 3eqtr4g 2793 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ×× 𝐶) = (𝐵 ×× 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 {cab 2705 ∃wrex 3067 ⟪caltop 35585 ×× caltxp 35586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-rex 3068 df-altxp 35588 |
This theorem is referenced by: (None) |
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