Theorem wl-2spsbbi 37519

Description: spsbbi 2073 applied twice. (Contributed by Wolf Lammen, 5-Aug-2023.)

Assertion

Ref	Expression
wl-2spsbbi	⊢ (∀𝑎∀𝑏(𝜑 ↔ 𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓))

Proof of Theorem wl-2spsbbi

Step	Hyp	Ref	Expression
1		alcom 2160	. 2 ⊢ (∀𝑎∀𝑏(𝜑 ↔ 𝜓) ↔ ∀𝑏∀𝑎(𝜑 ↔ 𝜓))
2		nfa1 2152	. . 3 ⊢ Ⅎ𝑏∀𝑏∀𝑎(𝜑 ↔ 𝜓)
3		nfa1 2152	. . . . 5 ⊢ Ⅎ𝑎∀𝑎(𝜑 ↔ 𝜓)
4		sp 2184	. . . . 5 ⊢ (∀𝑎(𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
5	3, 4	sbbid 2247	. . . 4 ⊢ (∀𝑎(𝜑 ↔ 𝜓) → ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜓))
6	5	sps 2186	. . 3 ⊢ (∀𝑏∀𝑎(𝜑 ↔ 𝜓) → ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜓))
7	2, 6	sbbid 2247	. 2 ⊢ (∀𝑏∀𝑎(𝜑 ↔ 𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓))
8	1, 7	sylbi 217	1 ⊢ (∀𝑎∀𝑏(𝜑 ↔ 𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-or 847  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by: (None)