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Mirrors > Home > ILE Home > Th. List > spv | GIF version |
Description: Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.) |
Ref | Expression |
---|---|
spv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spv | ⊢ (∀𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spv.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | biimpd 144 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
3 | 2 | spimv 1811 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 df-nf 1461 |
This theorem is referenced by: spvv 1907 cbvalvw 1919 chvarv 1937 ru 2962 nalset 4134 tfisi 4587 tfr1onlemsucfn 6341 tfr1onlemsucaccv 6342 tfr1onlembxssdm 6344 tfr1onlembfn 6345 tfr1onlemres 6350 tfri1dALT 6352 tfrcllemsucfn 6354 tfrcllemsucaccv 6355 tfrcllembxssdm 6357 tfrcllembfn 6358 tfrcllemres 6363 findcard2 6889 findcard2s 6890 bj-nalset 14650 |
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