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Mirrors > Home > MPE Home > Th. List > sbrimvlem | Structured version Visualization version GIF version |
Description: Common proof template for sbrimvw 2102 and sbrimv 2314. The hypothesis is an instance of 19.21 2207. (Contributed by Wolf Lammen, 29-Jan-2024.) |
Ref | Expression |
---|---|
sbrimvlem.1 | ⊢ (∀𝑥(𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓))) |
Ref | Expression |
---|---|
sbrimvlem | ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 2093 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) | |
2 | bi2.04 391 | . . . 4 ⊢ ((𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ (𝑥 = 𝑦 → (𝜑 → 𝜓))) | |
3 | 2 | albii 1820 | . . 3 ⊢ (∀𝑥(𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) |
4 | sbrimvlem.1 | . . 3 ⊢ (∀𝑥(𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓))) | |
5 | 1, 3, 4 | 3bitr2i 301 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓))) |
6 | sb6 2093 | . . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓)) | |
7 | 6 | imbi2i 338 | . 2 ⊢ ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓))) |
8 | 5, 7 | bitr4i 280 | 1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 |
This theorem is referenced by: sbrimvw 2102 sbrimv 2314 |
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