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Mirrors > Home > MPE Home > Th. List > Mathboxes > ac6sf2 | Structured version Visualization version GIF version |
Description: Alternate version of ac6 9904 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 2979 | . . . 4 ⊢ Ⅎ𝑧𝐵 | |
3 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | nfs1v 2160 | . . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑 | |
5 | sbequ12 2253 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑)) | |
6 | 1, 2, 3, 4, 5 | cbvrexfw 3440 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑) |
7 | 6 | ralbii 3167 | . 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 3554 | . . 3 ⊢ (𝑧 = (𝑓‘𝑥) → ([𝑧 / 𝑦]𝜑 ↔ 𝜓)) |
12 | 8, 11 | ac6s 9908 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
13 | 7, 12 | sylbi 219 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 Ⅎwnf 1784 [wsb 2069 ∈ wcel 2114 Ⅎwnfc 2963 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ⟶wf 6353 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-reg 9058 ax-inf2 9106 ax-ac2 9887 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-en 8512 df-r1 9195 df-rank 9196 df-card 9370 df-ac 9544 |
This theorem is referenced by: acunirnmpt2f 30408 |
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