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Mirrors > Home > MPE Home > Th. List > drsb1 | Structured version Visualization version GIF version |
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 2-Jun-1993.) (New usage is discouraged.) |
Ref | Expression |
---|---|
drsb1 | ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ1 2028 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | |
2 | 1 | sps 2178 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) |
3 | 2 | imbi1d 341 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑦 = 𝑧 → 𝜑))) |
4 | 2 | anbi1d 630 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧 ∧ 𝜑) ↔ (𝑦 = 𝑧 ∧ 𝜑))) |
5 | 4 | drex1 2440 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑧 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑))) |
6 | 3, 5 | anbi12d 631 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)) ↔ ((𝑦 = 𝑧 → 𝜑) ∧ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑)))) |
7 | dfsb1 2480 | . 2 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) | |
8 | dfsb1 2480 | . 2 ⊢ ([𝑧 / 𝑦]𝜑 ↔ ((𝑦 = 𝑧 → 𝜑) ∧ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑))) | |
9 | 6, 7, 8 | 3bitr4g 313 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 ∃wex 1781 [wsb 2067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-12 2171 ax-13 2371 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-nf 1786 df-sb 2068 |
This theorem is referenced by: sb2ae 2495 sbco3 2512 iotaeq 6505 |
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