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Mirrors > Home > MPE Home > Th. List > raldifeq | Structured version Visualization version GIF version |
Description: Equality theorem for restricted universal quantifier. (Contributed by Thierry Arnoux, 6-Jul-2019.) |
Ref | Expression |
---|---|
raldifeq.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
raldifeq.2 | ⊢ (𝜑 → ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓) |
Ref | Expression |
---|---|
raldifeq | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raldifeq.2 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓) | |
2 | 1 | biantrud 528 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓))) |
3 | ralunb 3992 | . . 3 ⊢ (∀𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝜓 ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓)) | |
4 | 2, 3 | syl6bbr 281 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝜓)) |
5 | raldifeq.1 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
6 | undif 4243 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) | |
7 | 5, 6 | sylib 210 | . . 3 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
8 | 7 | raleqdv 3327 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
9 | 4, 8 | bitrd 271 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∀wral 3089 ∖ cdif 3766 ∪ cun 3767 ⊆ wss 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 |
This theorem is referenced by: cantnfrescl 8823 rrxmet 23530 ntrneiel2 39166 ntrneik4w 39180 |
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