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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege61c | Structured version Visualization version GIF version |
Description: Lemma for frege65c 40629. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege59c.a | ⊢ 𝐴 ∈ 𝐵 |
Ref | Expression |
---|---|
frege61c | ⊢ (([𝐴 / 𝑥]𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege59c.a | . . 3 ⊢ 𝐴 ∈ 𝐵 | |
2 | 1 | frege58c 40622 | . 2 ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) |
3 | frege9 40513 | . 2 ⊢ ((∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) → (([𝐴 / 𝑥]𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ (([𝐴 / 𝑥]𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 ∈ wcel 2111 [wsbc 3720 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-frege1 40491 ax-frege2 40492 ax-frege8 40510 ax-frege58b 40602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-clab 2777 df-cleq 2791 df-clel 2870 df-sbc 3721 |
This theorem is referenced by: frege65c 40629 |
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