Theorem bds 13049

Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 13020; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 13020. (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 13047	. . 3 ⊢ BOUNDED {𝑥 ∣ 𝜑}
3		bds.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	cbvabv 2264	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
5	2, 4	bdceqi 13041	. 2 ⊢ BOUNDED {𝑦 ∣ 𝜓}
6	5	bdph 13048	1 ⊢ BOUNDED 𝜓

Syntax hints:   → wi 4   ↔ wb 104  {cab 2125  BOUNDED wbd 13010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bd0 13011  ax-bdsb 13020

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-bdc 13039

This theorem is referenced by: (None)