Theorem sbievg 2362

Description: Substitution applied to expressions linked by implicit substitution. The proof was part of a former cbvabw 2812 version. (Contributed by GG and WL, 26-Oct-2024.)

Hypotheses

Ref	Expression
sbievg.1	⊢ Ⅎ𝑦𝜑
sbievg.2	⊢ Ⅎ𝑥𝜓
sbievg.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem sbievg

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = 𝑤
2		sbievg.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
3	1, 2	nfim 1904	. . . . 5 ⊢ Ⅎ𝑦(𝑥 = 𝑤 → 𝜑)
4		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝑤
5		sbievg.2	. . . . . 6 ⊢ Ⅎ𝑥𝜓
6	4, 5	nfim 1904	. . . . 5 ⊢ Ⅎ𝑥(𝑦 = 𝑤 → 𝜓)
7		equequ1 2033	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑤 ↔ 𝑦 = 𝑤))
8		sbievg.3	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
9	7, 8	imbi12d 348	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝑤 → 𝜑) ↔ (𝑦 = 𝑤 → 𝜓)))
10	3, 6, 9	cbvalv1 2341	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑤 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑤 → 𝜓))
11	10	imbi2i 339	. . 3 ⊢ ((𝑤 = 𝑧 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) ↔ (𝑤 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜓)))
12	11	albii 1827	. 2 ⊢ (∀𝑤(𝑤 = 𝑧 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) ↔ ∀𝑤(𝑤 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜓)))
13		df-sb 2071	. 2 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑤(𝑤 = 𝑧 → ∀𝑥(𝑥 = 𝑤 → 𝜑)))
14		df-sb 2071	. 2 ⊢ ([𝑧 / 𝑦]𝜓 ↔ ∀𝑤(𝑤 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜓)))
15	12, 13, 14	3bitr4i 306	1 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541  Ⅎwnf 1791  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-11 2158  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792  df-sb 2071

This theorem is referenced by:  cbvabw  2812