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| Mirrors > Home > MPE Home > Th. List > sbc19.21g | Structured version Visualization version GIF version | ||
| Description: Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.) |
| Ref | Expression |
|---|---|
| sbcgf.1 | ⊢ Ⅎ𝑥𝜑 |
| Ref | Expression |
|---|---|
| sbc19.21g | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbcimg 3837 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) | |
| 2 | sbcgf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 3 | 2 | sbcgf 3861 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
| 4 | 3 | imbi1d 341 | . 2 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓))) |
| 5 | 1, 4 | bitrd 279 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 Ⅎwnf 1783 ∈ wcel 2108 [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-10 2141 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 |
| This theorem is referenced by: bnj121 34884 bnj124 34885 bnj130 34888 bnj207 34895 bnj611 34932 bnj1000 34955 |
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