Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ceqsralv | Structured version Visualization version GIF version |
Description: Restricted quantifier version of ceqsalv 3457. (Contributed by NM, 21-Jun-2013.) Avoid ax-9 2118, ax-12 2173, ax-ext 2709. (Revised by SN, 8-Sep-2024.) |
Ref | Expression |
---|---|
ceqsralv.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsralv | ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ceqsralv.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | pm5.74i 270 | . . 3 ⊢ ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓)) |
3 | 2 | ralbii 3090 | . 2 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜓)) |
4 | r19.23v 3207 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓)) | |
5 | risset 3193 | . . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴) | |
6 | pm5.5 361 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → ((∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓) ↔ 𝜓)) | |
7 | 5, 6 | sylbi 216 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ((∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓) ↔ 𝜓)) |
8 | 4, 7 | syl5bb 282 | . 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜓) ↔ 𝜓)) |
9 | 3, 8 | syl5bb 282 | 1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1784 df-clel 2817 df-ral 3068 df-rex 3069 |
This theorem is referenced by: eqreu 3659 sqrt2irr 15886 acsfn 17285 ovolgelb 24549 fsuppind 40202 |
Copyright terms: Public domain | W3C validator |