Theorem bds 10642

Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 10613; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 10613. (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 10640	. . 3 ⊢ BOUNDED {𝑥 ∣ 𝜑}
3		bds.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	cbvabv 2202	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
5	2, 4	bdceqi 10634	. 2 ⊢ BOUNDED {𝑦 ∣ 𝜓}
6	5	bdph 10641	1 ⊢ BOUNDED 𝜓

Syntax hints:   → wi 4   ↔ wb 103  {cab 2067  BOUNDED wbd 10603

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bd0 10604  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-bdc 10632

This theorem is referenced by: (None)