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Mathbox for Jonathan Ben-Naim |
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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 3842 | . 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 223 | . . 3 ⊢ ([𝑀 / 𝑛]𝜑 ↔ 𝜑′) |
4 | bnj206.2 | . . . 4 ⊢ (𝜓′ ↔ [𝑀 / 𝑛]𝜓) | |
5 | 4 | bicomi 223 | . . 3 ⊢ ([𝑀 / 𝑛]𝜓 ↔ 𝜓′) |
6 | bnj206.3 | . . . 4 ⊢ (𝜒′ ↔ [𝑀 / 𝑛]𝜒) | |
7 | 6 | bicomi 223 | . . 3 ⊢ ([𝑀 / 𝑛]𝜒 ↔ 𝜒′) |
8 | 3, 5, 7 | 3anbi123i 1152 | . 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 205 ∧ w3a 1084 ∈ wcel 2098 Vcvv 3468 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-sbc 3773 |
This theorem is referenced by: bnj124 34411 bnj207 34421 |
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