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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > riotaeqbidva | Structured version Visualization version GIF version |
Description: Equivalent wff's yield equal restricted definition binders (deduction form). (raleqbidva 3328 analog.) (Contributed by Thierry Arnoux, 29-Jan-2025.) |
Ref | Expression |
---|---|
riotaeqbidva.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
riotaeqbidva.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
riotaeqbidva | ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotaeqbidva.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
2 | 1 | riotabidva 7380 | . 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒)) |
3 | riotaeqbidva.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | riotaeqdv 7361 | . 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥 ∈ 𝐵 𝜒)) |
5 | 2, 4 | eqtrd 2773 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ℩crio 7359 |
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-v 3477 df-in 3954 df-ss 3964 df-uni 4908 df-iota 6492 df-riota 7360 |
This theorem is referenced by: ressply1invg 32605 |
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