Theorem bdsbc 9978

Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 9979. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdcsbc.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdsbc	⊢ BOUNDED [𝑦 / 𝑥]𝜑

Proof of Theorem bdsbc

Step	Hyp	Ref	Expression
1		bdcsbc.1	. . 3 ⊢ BOUNDED 𝜑
2	1	ax-bdsb 9942	. 2 ⊢ BOUNDED [𝑦 / 𝑥]𝜑
3		sbsbc 2768	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	bd0 9944	1 ⊢ BOUNDED [𝑦 / 𝑥]𝜑

Syntax hints:  [wsb 1645  [wsbc 2764  BOUNDED wbd 9932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933  ax-bdsb 9942

This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765

This theorem is referenced by:  bdccsb  9980