Theorem sbievw2 2104

Description: sbievw 2100 applied twice, avoiding a DV condition on 𝑥, 𝑦. Based on proofs by Wolf Lammen. (Contributed by Steven Nguyen, 29-Jul-2023.)

Hypotheses

Ref	Expression
sbievw2.1	⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))
sbievw2.2	⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓))

Assertion

Ref	Expression
sbievw2	⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Distinct variable groups:   𝑥,𝑤   𝑦,𝑤   𝜑,𝑤   𝜓,𝑤   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦,𝑤)

Proof of Theorem sbievw2

Step	Hyp	Ref	Expression
1		sbcom3vv 2103	. . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑦 / 𝑥]𝜑)
2		sbievw2.1	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))
3	2	sbievw 2100	. . . 4 ⊢ ([𝑤 / 𝑥]𝜑 ↔ 𝜒)
4	3	sbbii 2081	. . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤]𝜒)
5		sbv 2095	. . 3 ⊢ ([𝑦 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
6	1, 4, 5	3bitr3i 304	. 2 ⊢ ([𝑦 / 𝑤]𝜒 ↔ [𝑦 / 𝑥]𝜑)
7		sbievw2.2	. . 3 ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓))
8	7	sbievw 2100	. 2 ⊢ ([𝑦 / 𝑤]𝜒 ↔ 𝜓)
9	6, 8	bitr3i 280	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 209  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070

This theorem is referenced by:  sbco2vv  2105  equsb3  2106  equsb3r  2107  elsb3  2119  elsb4  2127  eqsb3  2916  clelsb3  2917  sbss  4420