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Mirrors > Home > MPE Home > Th. List > ralab | Structured version Visualization version GIF version |
Description: Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
ralab.1 | ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ralab | ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3143 | . 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒)) | |
2 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | ralab.1 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | elab 3666 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓) |
5 | 4 | imbi1i 352 | . . 3 ⊢ ((𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ (𝜓 → 𝜒)) |
6 | 5 | albii 1816 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ ∀𝑥(𝜓 → 𝜒)) |
7 | 1, 6 | bitri 277 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 ∈ wcel 2110 {cab 2799 ∀wral 3138 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 |
This theorem is referenced by: ralrnmpo 7288 funcnvuni 7635 kardex 9322 karden 9323 fimaxre3 11586 ptcnp 22229 ptrescn 22246 itg2leub 24334 nmoubi 28548 nmopub 29684 nmfnleub 29701 nmcexi 29802 mblfinlem3 34930 ismblfin 34932 itg2addnc 34945 hbtlem2 39722 |
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