Theorem bj-sbievw 34171

Description: Lemma for substitution. Closed form of equsalvw 2010 and sbievw 2103. (Contributed by BJ, 23-Jul-2023.)

Assertion

Ref	Expression
bj-sbievw	⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))

Distinct variable groups:   𝜓,𝑥   𝑥,𝑦

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

Proof of Theorem bj-sbievw

Step	Hyp	Ref	Expression
1		sb6 2093	. 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)))
2		bj-sblem 34168	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)))
3		sb6 2093	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
4		ax6ev 1972	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
5	4	a1bi 365	. . 3 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓))
6	2, 3, 5	3bitr4g 316	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
7	1, 6	sylbi 219	1 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535  ∃wex 1780  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070

This theorem is referenced by: (None)