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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 3139 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝑧 = ⟪𝑥, 𝑦⟫)) | |
2 | 1 | abbidv 2741 | . 2 ⊢ (𝐴 = 𝐵 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑧 = ⟪𝑥, 𝑦⟫} = {𝑧 ∣ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝑧 = ⟪𝑥, 𝑦⟫}) |
3 | df-altxp 32066 | . 2 ⊢ (𝐴 ×× 𝐶) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑧 = ⟪𝑥, 𝑦⟫} | |
4 | df-altxp 32066 | . 2 ⊢ (𝐵 ×× 𝐶) = {𝑧 ∣ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝑧 = ⟪𝑥, 𝑦⟫} | |
5 | 2, 3, 4 | 3eqtr4g 2681 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ×× 𝐶) = (𝐵 ×× 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 {cab 2608 ∃wrex 2913 ⟪caltop 32063 ×× caltxp 32064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-altxp 32066 |
This theorem is referenced by: (None) |
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