Theorem sbhypf 2779

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)

Hypotheses

Ref	Expression
sbhypf.1	⊢ Ⅎ𝑥𝜓
sbhypf.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbhypf	⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦

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

Proof of Theorem sbhypf

Step	Hyp	Ref	Expression
1		vex 2733	. . 3 ⊢ 𝑦 ∈ V
2		eqeq1 2177	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
3	1, 2	ceqsexv 2769	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
4		nfs1v 1932	. . . 4 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
5		sbhypf.1	. . . 4 ⊢ Ⅎ𝑥𝜓
6	4, 5	nfbi 1582	. . 3 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ↔ 𝜓)
7		sbequ12 1764	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8	7	bicomd 140	. . . 4 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
9		sbhypf.2	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
10	8, 9	sylan9bb 459	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
11	6, 10	exlimi 1587	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
12	3, 11	sylbir 134	1 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  Ⅎwnf 1453  ∃wex 1485  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  mob2  2910  cbvmptf  4083  tfisi  4571  ralxpf  4757  rexxpf  4758  nn0ind-raph  9329