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| Mirrors > Home > MPE Home > Th. List > ssrexf | Structured version Visualization version GIF version | ||
| Description: Restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
| Ref | Expression |
|---|---|
| ssrexf.1 | ⊢ Ⅎ𝑥𝐴 |
| ssrexf.2 | ⊢ Ⅎ𝑥𝐵 |
| Ref | Expression |
|---|---|
| ssrexf | ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssrexf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 2 | ssrexf.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 3 | 1, 2 | nfss 3976 | . . 3 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵 |
| 4 | ssel 3977 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
| 5 | 4 | anim1d 611 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) |
| 6 | 3, 5 | eximd 2216 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) |
| 7 | df-rex 3071 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 8 | df-rex 3071 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
| 9 | 6, 7, 8 | 3imtr4g 296 | 1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 Ⅎwnfc 2890 ∃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 ax-6 1967 ax-7 2007 ax-8 2110 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-ss 3968 |
| This theorem is referenced by: iunxdif3 5095 stoweidlem34 46049 |
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