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Mirrors > Home > MPE Home > Th. List > ralsngf | Structured version Visualization version GIF version |
Description: Restricted universal quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by AV, 3-Apr-2023.) |
Ref | Expression |
---|---|
rexsngf.1 | ⊢ Ⅎ𝑥𝜓 |
rexsngf.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ralsngf | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralsnsg 4677 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | rexsngf.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | rexsngf.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | sbciegf 3818 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
5 | 1, 4 | bitrd 278 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ∀wral 3058 [wsbc 3778 {csn 4632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2166 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-v 3475 df-sbc 3779 df-sn 4633 |
This theorem is referenced by: reusngf 4681 ralsngOLD 4686 rexreusng 4688 ralprgf 4701 |
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