Theorem bj-ssbssblem 32649

Description: Composition of two substitutions with a fresh intermediate variable. Remark: does not seem useful. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssbssblem	⊢ ([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑)

Distinct variable groups:   𝑦,𝑡   𝑥,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ssbssblem

Step	Hyp	Ref	Expression
1		bj-ssb1 32633	. 2 ⊢ ([𝑡/𝑦]b∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		bj-ssb1 32633	. . 3 ⊢ ([𝑦/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	2	bj-ssbbii 32624	. 2 ⊢ ([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑦]b∀𝑥(𝑥 = 𝑦 → 𝜑))
4		df-ssb 32620	. 2 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	1, 3, 4	3bitr4i 292	1 ⊢ ([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑)

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  [wssb 32619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-11 2034

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ssb 32620

This theorem is referenced by: (None)