Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ralvw | Structured version Visualization version GIF version |
Description: A weak version of ralv 3219 not using ax-ext 2602 (nor df-cleq 2615, df-clel 2618, df-v 3202), but using ax-13 2246. For the sake of illustration, the next theorem bj-rexvwv 32866, a weak version of rexv 3220, has a dv condition and avoids dependency on ax-13 2246, while the analogues for reuv 3221 and rmov 3222 are not proved. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ralvw.1 | ⊢ 𝜓 |
Ref | Expression |
---|---|
bj-ralvw | ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2917 | . 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑)) | |
2 | bj-ralvw.1 | . . . . 5 ⊢ 𝜓 | |
3 | 2 | bj-vexw 32855 | . . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓} |
4 | 3 | a1bi 352 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑)) |
5 | 4 | albii 1747 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑)) |
6 | 1, 5 | bitr4i 267 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 ∈ wcel 1990 {cab 2608 ∀wral 2912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-sb 1881 df-clab 2609 df-ral 2917 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |