Theorem bnd 9844

Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 5236), derived from the Collection Principle cp 9843. 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 9843	. . 3 ⊢ ∃𝑤∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑)
2		ralim 3101	. . 3 ⊢ (∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑) → (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑))
3	1, 2	eximii 1856	. 2 ⊢ ∃𝑤(∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)
4	3	19.37iv 1967	1 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Syntax hints:   → wi 4  ∃wex 1798  ∀wral 3075  ∃wrex 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-reg 9534  ax-inf2 9590

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-om 7842  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-r1 9716  df-rank 9717

This theorem is referenced by:  bnd2  9845