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Mirrors > Home > NFE Home > Th. List > sbcimdv | GIF version |
Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.) |
Ref | Expression |
---|---|
sbcimdv.1 | ⊢ (φ → (ψ → χ)) |
Ref | Expression |
---|---|
sbcimdv | ⊢ ((φ ∧ A ∈ V) → ([̣A / x]̣ψ → [̣A / x]̣χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcimdv.1 | . . . . 5 ⊢ (φ → (ψ → χ)) | |
2 | 1 | alrimiv 1631 | . . . 4 ⊢ (φ → ∀x(ψ → χ)) |
3 | spsbc 3058 | . . . 4 ⊢ (A ∈ V → (∀x(ψ → χ) → [̣A / x]̣(ψ → χ))) | |
4 | 2, 3 | syl5 28 | . . 3 ⊢ (A ∈ V → (φ → [̣A / x]̣(ψ → χ))) |
5 | sbcimg 3087 | . . 3 ⊢ (A ∈ V → ([̣A / x]̣(ψ → χ) ↔ ([̣A / x]̣ψ → [̣A / x]̣χ))) | |
6 | 4, 5 | sylibd 205 | . 2 ⊢ (A ∈ V → (φ → ([̣A / x]̣ψ → [̣A / x]̣χ))) |
7 | 6 | impcom 419 | 1 ⊢ ((φ ∧ A ∈ V) → ([̣A / x]̣ψ → [̣A / x]̣χ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∀wal 1540 ∈ wcel 1710 [̣wsbc 3046 |
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-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 |
This theorem is referenced by: (None) |
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