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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2rspcedvdw | Structured version Visualization version GIF version |
Description: Double application of rspcedvdw 39731. (Contributed by SN, 24-Aug-2024.) |
Ref | Expression |
---|---|
2rspcedvdw.1 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) |
2rspcedvdw.2 | ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) |
2rspcedvdw.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
2rspcedvdw.b | ⊢ (𝜑 → 𝐵 ∈ 𝑌) |
2rspcedvdw.3 | ⊢ (𝜑 → 𝜃) |
Ref | Expression |
---|---|
2rspcedvdw | ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2rspcedvdw.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
2 | 2rspcedvdw.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑌) | |
3 | 2rspcedvdw.3 | . 2 ⊢ (𝜑 → 𝜃) | |
4 | 2rspcedvdw.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) | |
5 | 2rspcedvdw.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) | |
6 | 4, 5 | rspc2ev 3555 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝜃) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓) |
7 | 1, 2, 3, 6 | syl3anc 1368 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃wrex 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2729 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1086 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-ral 3075 df-rex 3076 |
This theorem is referenced by: flt4lem7 40023 nna4b4nsq 40024 |
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