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| Description: Two existential quantifications restricted to a subclass. (Contributed by AV, 11-Mar-2023.) | 
| Ref | Expression | 
|---|---|
| ss2rexv | ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssrexv 4053 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦 ∈ 𝐴 𝜑 → ∃𝑦 ∈ 𝐵 𝜑)) | |
| 2 | 1 | reximdv 3170 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)) | 
| 3 | ssrexv 4053 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝜑)) | |
| 4 | 2, 3 | syld 47 | 1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∃wrex 3070 ⊆ wss 3951 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-rex 3071 df-ss 3968 | 
| This theorem is referenced by: prpair 47488 grtriprop 47908 | 
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