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Mirrors > Home > MPE Home > Th. List > alexeqg | Structured version Visualization version GIF version |
Description: Two ways to express substitution of 𝐴 for 𝑥 in 𝜑. This is the analogue for classes of sb56 2274. (Contributed by NM, 2-Mar-1995.) (Revised by BJ, 27-Apr-2019.) |
Ref | Expression |
---|---|
alexeqg | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2810 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
2 | 1 | anbi1d 632 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑))) |
3 | 2 | exbidv 1922 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
4 | 1 | imbi1d 345 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜑))) |
5 | 4 | albidv 1921 | . . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) |
6 | sb56 2274 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
7 | 3, 5, 6 | vtoclbg 3517 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) |
8 | 7 | bicomd 226 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 ∃wex 1781 ∈ wcel 2111 |
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-cleq 2791 df-clel 2870 |
This theorem is referenced by: ceqex 3593 sbc6g 3749 |
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