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Mirrors > Home > MPE Home > Th. List > sbc6 | Structured version Visualization version GIF version |
Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
Ref | Expression |
---|---|
sbc6.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
sbc6 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc6.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sbc6g 3461 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 ∈ wcel 1990 Vcvv 3200 [wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 df-sbc 3436 |
This theorem is referenced by: intab 4507 2sbc6g 38616 |
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