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Mirrors > Home > MPE Home > Th. List > r19.26m | Structured version Visualization version GIF version |
Description: Version of 19.26 1878 and r19.26 3092 with restricted quantifiers ranging over different classes. (Contributed by NM, 22-Feb-2004.) |
Ref | Expression |
---|---|
r19.26m | ⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝜑) ∧ (𝑥 ∈ 𝐵 → 𝜓)) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.26 1878 | . 2 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝜑) ∧ (𝑥 ∈ 𝐵 → 𝜓)) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))) | |
2 | df-ral 3066 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
3 | df-ral 3066 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) | |
4 | 2, 3 | anbi12i 630 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜓) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))) |
5 | 1, 4 | bitr4i 281 | 1 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝜑) ∧ (𝑥 ∈ 𝐵 → 𝜓)) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 ∈ wcel 2110 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ral 3066 |
This theorem is referenced by: (None) |
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