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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 1888 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | |
2 | sbc6g 3834 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | |
3 | 2 | bicomd 223 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ [𝐴 / 𝑥]𝜑)) |
4 | 3 | anbi2d 629 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑))) |
5 | 1, 4 | bitrid 283 | 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 206 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2108 [wsbc 3804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-sbc 3805 |
This theorem is referenced by: pm14.122c 44393 |
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