![]() |
Mathbox for Giovanni Mascellani |
< Previous
Next >
Nearby theorems |
|
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 3101 | . 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 3351 | . 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 1537 Ⅎwnfc 2888 ∀wral 3059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |