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Mirrors > Home > NFE Home > Th. List > sbcco3g | GIF version |
Description: Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) |
Ref | Expression |
---|---|
sbcco3g.1 | ⊢ (x = A → B = C) |
Ref | Expression |
---|---|
sbcco3g | ⊢ (A ∈ V → ([̣A / x]̣[̣B / y]̣φ ↔ [̣C / y]̣φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcnestg 3185 | . 2 ⊢ (A ∈ V → ([̣A / x]̣[̣B / y]̣φ ↔ [̣[A / x]B / y]̣φ)) | |
2 | elex 2867 | . . 3 ⊢ (A ∈ V → A ∈ V) | |
3 | nfcvd 2490 | . . . 4 ⊢ (A ∈ V → ℲxC) | |
4 | sbcco3g.1 | . . . 4 ⊢ (x = A → B = C) | |
5 | 3, 4 | csbiegf 3176 | . . 3 ⊢ (A ∈ V → [A / x]B = C) |
6 | dfsbcq 3048 | . . 3 ⊢ ([A / x]B = C → ([̣[A / x]B / y]̣φ ↔ [̣C / y]̣φ)) | |
7 | 2, 5, 6 | 3syl 18 | . 2 ⊢ (A ∈ V → ([̣[A / x]B / y]̣φ ↔ [̣C / y]̣φ)) |
8 | 1, 7 | bitrd 244 | 1 ⊢ (A ∈ V → ([̣A / x]̣[̣B / y]̣φ ↔ [̣C / y]̣φ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 Vcvv 2859 [̣wsbc 3046 [csb 3136 |
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-v 2861 df-sbc 3047 df-csb 3137 |
This theorem is referenced by: sbcco3gOLD 3192 |
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