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Mirrors > Home > MPE Home > Th. List > 2stdpc4 | Structured version Visualization version GIF version |
Description: A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2071 for the analogous single specialization. See 2sp 2179 for another double specialization. (Contributed by Andrew Salmon, 24-May-2011.) (Revised by BJ, 21-Oct-2018.) |
Ref | Expression |
---|---|
2stdpc4 | ⊢ (∀𝑥∀𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdpc4 2071 | . . 3 ⊢ (∀𝑦𝜑 → [𝑤 / 𝑦]𝜑) | |
2 | 1 | alimi 1814 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥[𝑤 / 𝑦]𝜑) |
3 | stdpc4 2071 | . 2 ⊢ (∀𝑥[𝑤 / 𝑦]𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | |
4 | 2, 3 | syl 17 | 1 ⊢ (∀𝑥∀𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 [wsb 2067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 |
This theorem depends on definitions: df-bi 206 df-sb 2068 |
This theorem is referenced by: ax11-pm2 35019 pm11.11 41992 |
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