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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ac6sf2 | Structured version Visualization version GIF version |
Description: Alternate version of ac6 9891 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 2955 | . . . 4 ⊢ Ⅎ𝑧𝐵 | |
3 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | nfs1v 2157 | . . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑 | |
5 | sbequ12 2250 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑)) | |
6 | 1, 2, 3, 4, 5 | cbvrexfw 3384 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑) |
7 | 6 | ralbii 3133 | . 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 3500 | . . 3 ⊢ (𝑧 = (𝑓‘𝑥) → ([𝑧 / 𝑦]𝜑 ↔ 𝜓)) |
12 | 8, 11 | ac6s 9895 | . 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 1538 ∃wex 1781 Ⅎwnf 1785 [wsb 2069 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ⟶wf 6320 ‘cfv 6324 |
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-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 ax-ac2 9874 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-en 8493 df-r1 9177 df-rank 9178 df-card 9352 df-ac 9527 |
This theorem is referenced by: acunirnmpt2f 30424 |
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