Theorem bnd 9052

Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 7413), derived from the Collection Principle cp 9051. 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. (Contributed by NM, 17-Oct-2004.)

Assertion

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

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bnd

Step	Hyp	Ref	Expression
1		cp 9051	. . 3 ⊢ ∃𝑤∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑)
2		ralim 3129	. . 3 ⊢ (∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑) → (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑))
3	1, 2	eximii 1880	. 2 ⊢ ∃𝑤(∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)
4	3	19.37iv 1991	1 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Syntax hints:   → wi 4  ∃wex 1823  ∀wral 3089  ∃wrex 3090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-reg 8786  ax-inf2 8835

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-r1 8924  df-rank 8925

This theorem is referenced by:  bnd2  9053