Theorem bnd 3548

Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 2744), derived from the Collection Principle cp 3547. Its strength lies in the rather profound fact that φ(x, y) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom.

Assertion

Ref	Expression
bnd	⊢ (∀x ∈ z ∃yφ → ∃w∀x ∈ z ∃y ∈ w φ)

Distinct variable group(s):   φ,z,w   x,y,z,w

Proof of Theorem bnd

Step	Hyp	Ref	Expression
1		cp 3547	. . 3 ⊢ ∃w∀x ∈ z (∃yφ → ∃y ∈ w φ)
2		r19.20 1251	. . . 4 ⊢ (∀x ∈ z (∃yφ → ∃y ∈ w φ) → (∀x ∈ z ∃yφ → ∀x ∈ z ∃y ∈ w φ))
3	2	19.22i 723	. . 3 ⊢ (∃w∀x ∈ z (∃yφ → ∃y ∈ w φ) → ∃w(∀x ∈ z ∃yφ → ∀x ∈ z ∃y ∈ w φ))
4	1, 3	ax-mp 6	. 2 ⊢ ∃w(∀x ∈ z ∃yφ → ∀x ∈ z ∃y ∈ w φ)
5		19.37v 961	. 2 ⊢ (∃w(∀x ∈ z ∃yφ → ∀x ∈ z ∃y ∈ w φ) ↔ (∀x ∈ z ∃yφ → ∃w∀x ∈ z ∃y ∈ w φ))
6	4, 5	mpbi 164	1 ⊢ (∀x ∈ z ∃yφ → ∃w∀x ∈ z ∃y ∈ w φ)

Syntax hints:   → wi 2  ∃wex 678  ∀wral 1201  ∃wrex 1202

This theorem is referenced by:  bnd2 3549

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-iin 1997  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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