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Mirrors > Home > MPE Home > Th. List > riotabiia | Structured version Visualization version GIF version |
Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3435 analog.) (Contributed by NM, 16-Jan-2012.) |
Ref | Expression |
---|---|
riotabiia.1 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
riotabiia | ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . 2 ⊢ V = V | |
2 | riotabiia.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | adantl 482 | . . 3 ⊢ ((V = V ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓)) |
4 | 3 | riotabidva 7369 | . 2 ⊢ (V = V → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓)) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ℩crio 7348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-in 3951 df-ss 3961 df-uni 4902 df-iota 6484 df-riota 7349 |
This theorem is referenced by: riotaxfrd 7384 lubfval 18285 glbfval 18298 odulub 18342 oduglb 18344 cnlnadjlem5 31187 cdj3lem3 31554 cdj3lem3b 31556 lshpkrlem1 37785 cdleme25cv 39034 cdlemk35 39588 |
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