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Mirrors > Home > NFE Home > Th. List > ralsng | 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 |
---|---|
ralsng.1 | ⊢ (x = A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
ralsng | ⊢ (A ∈ V → (∀x ∈ {A}φ ↔ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralsns 3763 | . 2 ⊢ (A ∈ V → (∀x ∈ {A}φ ↔ [̣A / x]̣φ)) | |
2 | ralsng.1 | . . 3 ⊢ (x = A → (φ ↔ ψ)) | |
3 | 2 | sbcieg 3078 | . 2 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ ψ)) |
4 | 1, 3 | bitrd 244 | 1 ⊢ (A ∈ V → (∀x ∈ {A}φ ↔ ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 ∀wral 2614 [̣wsbc 3046 {csn 3737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-v 2861 df-sbc 3047 df-sn 3741 |
This theorem is referenced by: ralsn 3767 ralprg 3775 raltpg 3777 ralunsn 3879 iinxsng 4042 |
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