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| Mirrors > Home > MPE Home > Th. List > 2ralsng | Structured version Visualization version GIF version | ||
| Description: Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.) | 
| Ref | Expression | 
|---|---|
| ralsng.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| 2ralsng.1 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| 2ralsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ralsng.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 2 | 1 | ralbidv 3177 | . . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓)) | 
| 3 | 2 | ralsng 4674 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓)) | 
| 4 | 2ralsng.1 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
| 5 | 4 | ralsng 4674 | . 2 ⊢ (𝐵 ∈ 𝑊 → (∀𝑦 ∈ {𝐵}𝜓 ↔ 𝜒)) | 
| 6 | 3, 5 | sylan9bb 509 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 {csn 4625 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-v 3481 df-sn 4626 | 
| This theorem is referenced by: c0snmgmhm 20463 zrrnghm 20537 mat1ghm 22490 mat1mhm 22491 f1resfz0f1d 35120 | 
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