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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ralbi12f | Structured version Visualization version GIF version | ||
| Description: Equality deduction for restricted universal quantification. (Contributed by Giovanni Mascellani, 10-Apr-2018.) | 
| Ref | Expression | 
|---|---|
| ralbi12f.1 | ⊢ Ⅎ𝑥𝐴 | 
| ralbi12f.2 | ⊢ Ⅎ𝑥𝐵 | 
| Ref | Expression | 
|---|---|
| ralbi12f | ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ralbi 3103 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)) | |
| 2 | ralbi12f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 3 | ralbi12f.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
| 4 | 2, 3 | raleqf 3353 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) | 
| 5 | 1, 4 | sylan9bbr 510 | 1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnfc 2890 ∀wral 3061 | 
| 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-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 | 
| This theorem is referenced by: (None) | 
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