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Mirrors > Home > MPE Home > Th. List > spsbbiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of spsbbi 2077 as of 6-Jul-2023. Specialization of biconditional. (Contributed by NM, 2-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spsbbiOLD | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdpc4 2072 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → [𝑦 / 𝑥](𝜑 ↔ 𝜓)) | |
2 | sbbi 2316 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) | |
3 | 1, 2 | sylib 220 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1534 [wsb 2068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 df-sb 2069 |
This theorem is referenced by: (None) |
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