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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj206 | Structured version Visualization version GIF version | ||
| Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bnj206.1 | ⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑) |
| bnj206.2 | ⊢ (𝜓′ ↔ [𝑀 / 𝑛]𝜓) |
| bnj206.3 | ⊢ (𝜒′ ↔ [𝑀 / 𝑛]𝜒) |
| bnj206.4 | ⊢ 𝑀 ∈ V |
| Ref | Expression |
|---|---|
| bnj206 | ⊢ ([𝑀 / 𝑛](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbc3an 3855 | . 2 ⊢ ([𝑀 / 𝑛](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ([𝑀 / 𝑛]𝜑 ∧ [𝑀 / 𝑛]𝜓 ∧ [𝑀 / 𝑛]𝜒)) | |
| 2 | bnj206.1 | . . . 4 ⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑) | |
| 3 | 2 | bicomi 224 | . . 3 ⊢ ([𝑀 / 𝑛]𝜑 ↔ 𝜑′) |
| 4 | bnj206.2 | . . . 4 ⊢ (𝜓′ ↔ [𝑀 / 𝑛]𝜓) | |
| 5 | 4 | bicomi 224 | . . 3 ⊢ ([𝑀 / 𝑛]𝜓 ↔ 𝜓′) |
| 6 | bnj206.3 | . . . 4 ⊢ (𝜒′ ↔ [𝑀 / 𝑛]𝜒) | |
| 7 | 6 | bicomi 224 | . . 3 ⊢ ([𝑀 / 𝑛]𝜒 ↔ 𝜒′) |
| 8 | 3, 5, 7 | 3anbi123i 1156 | . 2 ⊢ (([𝑀 / 𝑛]𝜑 ∧ [𝑀 / 𝑛]𝜓 ∧ [𝑀 / 𝑛]𝜒) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′)) |
| 9 | 1, 8 | bitri 275 | 1 ⊢ ([𝑀 / 𝑛](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ w3a 1087 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sbc 3789 |
| This theorem is referenced by: bnj124 34885 bnj207 34895 |
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