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Mirrors > Home > MPE Home > Th. List > sbievg | Structured version Visualization version GIF version |
Description: Substitution applied to expressions linked by implicit substitution. The proof was part of a former cbvabw 2812 version. (Contributed by GG and WL, 26-Oct-2024.) |
Ref | Expression |
---|---|
sbievg.1 | ⊢ Ⅎ𝑦𝜑 |
sbievg.2 | ⊢ Ⅎ𝑥𝜓 |
sbievg.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbievg | ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1922 | . . . . . 6 ⊢ Ⅎ𝑦 𝑥 = 𝑤 | |
2 | sbievg.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
3 | 1, 2 | nfim 1904 | . . . . 5 ⊢ Ⅎ𝑦(𝑥 = 𝑤 → 𝜑) |
4 | nfv 1922 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝑤 | |
5 | sbievg.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
6 | 4, 5 | nfim 1904 | . . . . 5 ⊢ Ⅎ𝑥(𝑦 = 𝑤 → 𝜓) |
7 | equequ1 2033 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑤 ↔ 𝑦 = 𝑤)) | |
8 | sbievg.3 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
9 | 7, 8 | imbi12d 348 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝑤 → 𝜑) ↔ (𝑦 = 𝑤 → 𝜓))) |
10 | 3, 6, 9 | cbvalv1 2341 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑤 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑤 → 𝜓)) |
11 | 10 | imbi2i 339 | . . 3 ⊢ ((𝑤 = 𝑧 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) ↔ (𝑤 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜓))) |
12 | 11 | albii 1827 | . 2 ⊢ (∀𝑤(𝑤 = 𝑧 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) ↔ ∀𝑤(𝑤 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜓))) |
13 | df-sb 2071 | . 2 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑤(𝑤 = 𝑧 → ∀𝑥(𝑥 = 𝑤 → 𝜑))) | |
14 | df-sb 2071 | . 2 ⊢ ([𝑧 / 𝑦]𝜓 ↔ ∀𝑤(𝑤 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜓))) | |
15 | 12, 13, 14 | 3bitr4i 306 | 1 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 Ⅎwnf 1791 [wsb 2070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-nf 1792 df-sb 2071 |
This theorem is referenced by: cbvabw 2812 |
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