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Mirrors > Home > MPE Home > Th. List > ralsnsg | Structured version Visualization version GIF version |
Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
ralsnsg | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3066 | . . 3 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝜑)) | |
2 | velsn 4607 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
3 | 2 | imbi1i 350 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} → 𝜑) ↔ (𝑥 = 𝐴 → 𝜑)) |
4 | 3 | albii 1822 | . . 3 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
5 | 1, 4 | bitri 275 | . 2 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
6 | sbc6g 3774 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | |
7 | 5, 6 | bitr4id 290 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ∀wral 3065 [wsbc 3744 {csn 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-v 3450 df-sbc 3745 df-sn 4592 |
This theorem is referenced by: ralsngf 4637 ixpsnval 8845 ac6sfi 9238 rexfiuz 15239 prmind2 16568 finixpnum 36092 |
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