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Mirrors > Home > MPE Home > Th. List > Mathboxes > spcdvw | Structured version Visualization version GIF version |
Description: A version of spcdv 3592 where 𝜓 and 𝜒 are direct substitutions of each other. This theorem is useful because it does not require 𝜑 and 𝑥 to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.) |
Ref | Expression |
---|---|
spcdvw.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
spcdvw.2 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
spcdvw | ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcdvw.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) | |
2 | 1 | biimpd 231 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜒)) |
3 | 2 | ax-gen 1792 | . 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) |
4 | spcdvw.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
5 | nfv 1911 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
6 | nfcv 2977 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
7 | 5, 6 | spcimgft 3585 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜓 → 𝜒))) |
8 | 3, 4, 7 | mpsyl 68 | 1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 = wceq 1533 ∈ wcel 2110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 |
This theorem is referenced by: setrec1lem4 44787 |
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