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Mirrors > Home > MPE Home > Th. List > Mathboxes > ceqsralv2 | Structured version Visualization version GIF version |
Description: Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.) |
Ref | Expression |
---|---|
ceqsralv2.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsralv2 | ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ceqsralv2.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | notbid 307 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓)) |
3 | 2 | ceqsrexv2 31912 | . . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝜓)) |
4 | rexanali 3136 | . . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑)) | |
5 | annim 440 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝜓) ↔ ¬ (𝐴 ∈ 𝐵 → 𝜓)) | |
6 | 3, 4, 5 | 3bitr3i 290 | . 2 ⊢ (¬ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ ¬ (𝐴 ∈ 𝐵 → 𝜓)) |
7 | 6 | con4bii 310 | 1 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ∃wrex 3051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-12 2196 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-ral 3055 df-rex 3056 df-v 3342 |
This theorem is referenced by: (None) |
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