Theorem bds 15056

Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 15027; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 15027. (Contributed by BJ, 19-Nov-2019.)

Hypotheses

Ref	Expression
bds.bd	⊢ BOUNDED 𝜑
bds.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bds	⊢ BOUNDED 𝜓

Distinct variable groups:   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem bds

Step	Hyp	Ref	Expression
1		bds.bd	. . . 4 ⊢ BOUNDED 𝜑
2	1	bdcab 15054	. . 3 ⊢ BOUNDED {𝑥 ∣ 𝜑}
3		bds.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	cbvabv 2314	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
5	2, 4	bdceqi 15048	. 2 ⊢ BOUNDED {𝑦 ∣ 𝜓}
6	5	bdph 15055	1 ⊢ BOUNDED 𝜓

Syntax hints:   → wi 4   ↔ wb 105  {cab 2175  BOUNDED wbd 15017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15018  ax-bdsb 15027

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-bdc 15046

This theorem is referenced by: (None)