Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ceqsrexv2 | Structured version Visualization version GIF version |
Description: Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.) |
Ref | Expression |
---|---|
ceqsrexv2.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsrexv2 | ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ceqsrexv2.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | ceqsrexbv 3650 | 1 ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 |
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-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2814 df-clel 2893 df-rex 3144 |
This theorem is referenced by: ceqsralv2 32956 |
Copyright terms: Public domain | W3C validator |