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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 3767 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) | |
2 | sbcgf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | sbcgf 3791 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
4 | 3 | imbi1d 345 | . 2 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓))) |
5 | 1, 4 | bitrd 282 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 Ⅎwnf 1785 ∈ wcel 2111 [wsbc 3720 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-sbc 3721 |
This theorem is referenced by: bnj121 32252 bnj124 32253 bnj130 32256 bnj207 32263 bnj611 32300 bnj1000 32323 |
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