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Mirrors > Home > MPE Home > Th. List > Mathboxes > ac6sf2 | Structured version Visualization version GIF version |
Description: Alternate version of ac6 9980 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.) (Revised by Thierry Arnoux, 17-May-2020.) |
Ref | Expression |
---|---|
ac6sf2.y | ⊢ Ⅎ𝑦𝐵 |
ac6sf2.1 | ⊢ Ⅎ𝑦𝜓 |
ac6sf2.2 | ⊢ 𝐴 ∈ V |
ac6sf2.3 | ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ac6sf2 | ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ac6sf2.y | . . . 4 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑧𝐵 | |
3 | nfv 1921 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | nfs1v 2161 | . . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑 | |
5 | sbequ12 2253 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑)) | |
6 | 1, 2, 3, 4, 5 | cbvrexfw 3337 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑) |
7 | 6 | ralbii 3080 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑) |
8 | ac6sf2.2 | . . 3 ⊢ 𝐴 ∈ V | |
9 | ac6sf2.1 | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
10 | ac6sf2.3 | . . . 4 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) | |
11 | 9, 10 | sbhypf 3456 | . . 3 ⊢ (𝑧 = (𝑓‘𝑥) → ([𝑧 / 𝑦]𝜑 ↔ 𝜓)) |
12 | 8, 11 | ac6s 9984 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
13 | 7, 12 | sylbi 220 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∃wex 1786 Ⅎwnf 1790 [wsb 2074 ∈ wcel 2114 Ⅎwnfc 2879 ∀wral 3053 ∃wrex 3054 Vcvv 3398 ⟶wf 6335 ‘cfv 6339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-reg 9129 ax-inf2 9177 ax-ac2 9963 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-en 8556 df-r1 9266 df-rank 9267 df-card 9441 df-ac 9616 |
This theorem is referenced by: acunirnmpt2f 30573 |
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