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Mirrors > Home > ILE Home > Th. List > ralrimi | GIF version |
Description: Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.) |
Ref | Expression |
---|---|
ralrimi.1 | ⊢ Ⅎ𝑥𝜑 |
ralrimi.2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) |
Ref | Expression |
---|---|
ralrimi | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralrimi.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ralrimi.2 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) | |
3 | 1, 2 | alrimi 1522 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) |
4 | df-ral 2460 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
5 | 3, 4 | sylibr 134 | 1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1351 Ⅎwnf 1460 ∈ wcel 2148 ∀wral 2455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-4 1510 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-ral 2460 |
This theorem is referenced by: ralrimiv 2549 reximdai 2575 r19.12 2583 rexlimd 2591 rexlimd2 2592 r19.29af2 2617 r19.37 2629 ralidm 3523 iineq2d 3906 mpteq2da 4092 onintonm 4516 mpteqb 5606 fmptdf 5673 eusvobj2 5860 tfri3 6367 mapxpen 6847 fodjuomnilemdc 7141 cc3 7266 fimaxre2 11230 fprodcllemf 11616 fprodap0f 11639 fprodle 11643 zsupcllemstep 11940 bezoutlemmain 11993 bezoutlemzz 11997 exmidunben 12421 mulcncf 13984 limccnp2lem 14038 |
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