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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 3729 | . 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 209 ∀wal 1541 = wceq 1543 ∈ wcel 2110 Vcvv 3413 [wsbc 3699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3700 |
This theorem is referenced by: intab 4894 sbcop1 5376 2sbc6g 41714 sbcpr 44654 |
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