Theorem bds 14606

Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 14577; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 14577. (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 14604	. . 3 ⊢ BOUNDED {𝑥 ∣ 𝜑}
3		bds.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	cbvabv 2302	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
5	2, 4	bdceqi 14598	. 2 ⊢ BOUNDED {𝑦 ∣ 𝜓}
6	5	bdph 14605	1 ⊢ BOUNDED 𝜓

Syntax hints:   → wi 4   ↔ wb 105  {cab 2163  BOUNDED wbd 14567

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14568  ax-bdsb 14577

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-bdc 14596

This theorem is referenced by: (None)