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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm14.122a | Structured version Visualization version GIF version |
Description: Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
Ref | Expression |
---|---|
pm14.122a | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albiim 1890 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | |
2 | sbc6g 3801 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | |
3 | 2 | bicomd 225 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ [𝐴 / 𝑥]𝜑)) |
4 | 3 | anbi2d 630 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑))) |
5 | 1, 4 | syl5bb 285 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3496 df-sbc 3773 |
This theorem is referenced by: pm14.122c 40776 |
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