Theorem bnd 4723

Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 3614), derived from the Collection Principle cp 4722. Its strength lies in the rather profound fact that ph(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	|- (A.x e. z E.yph -> E.wA.x e. z E.y e. w ph)

Distinct variable groups:   ph,z,w   x,y,z,w

Proof of Theorem bnd

Step	Hyp	Ref	Expression
1		cp 4722	. . 3 |- E.wA.x e. z (E.yph -> E.y e. w ph)
2		r19.20 1702	. . . 4 |- (A.x e. z (E.yph -> E.y e. w ph) -> (A.x e. z E.yph -> A.x e. z E.y e. w ph))
3	2	19.22i 1040	. . 3 |- (E.wA.x e. z (E.yph -> E.y e. w ph) -> E.w(A.x e. z E.yph -> A.x e. z E.y e. w ph))
4	1, 3	ax-mp 7	. 2 |- E.w(A.x e. z E.yph -> A.x e. z E.y e. w ph)
5		19.37v 1303	. 2 |- (E.w(A.x e. z E.yph -> A.x e. z E.y e. w ph) <-> (A.x e. z E.yph -> E.wA.x e. z E.y e. w ph))
6	4, 5	mpbi 189	1 |- (A.x e. z E.yph -> E.wA.x e. z E.y e. w ph)

Syntax hints:   -> wi 3  E.wex 980  A.wral 1645  E.wrex 1646

This theorem is referenced by:  bnd2 4724

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-r1 4643  df-rank 4644

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