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Mirrors > Home > MPE Home > Th. List > rspcedeq2vd | Structured version Visualization version GIF version |
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3585 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.) |
Ref | Expression |
---|---|
rspcedeqvd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcedeqvd.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
rspcedeq2vd | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcedeqvd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | rspcedeqvd.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷) | |
3 | 2 | eqcomd 2739 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐶) |
4 | 3 | eqeq2d 2744 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐶 = 𝐷 ↔ 𝐶 = 𝐶)) |
5 | eqidd 2734 | . 2 ⊢ (𝜑 → 𝐶 = 𝐶) | |
6 | 1, 4, 5 | rspcedvd 3585 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 |
This theorem is referenced by: symgextfo 19212 smatvscl 21896 eucrctshift 29236 ntrclsneine0lem 42428 mogoldbblem 46002 sbgoldbwt 46059 sbgoldbo 46069 |
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