Theorem vtocld 3516

Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-10 2144, ax-11 2160, ax-12 2180. (Revised by SN, 2-Sep-2024.)

Hypotheses

Ref	Expression
vtocld.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
vtocld.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
vtocld.3	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
vtocld	⊢ (𝜑 → 𝜒)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem vtocld

Step	Hyp	Ref	Expression
1		vtocld.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		elisset 2813	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
4		vtocld.3	. . . 4 ⊢ (𝜑 → 𝜓)
5	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓)
6		vtocld.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
7	5, 6	mpbid 232	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜒)
8	3, 7	exlimddv 1936	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111

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 1968  ax-7 2009  ax-8 2113

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-clel 2806

This theorem is referenced by:  vtocl2d  3517  lmatfval  33822  lmatcl  33824  bj-elabd2ALT  36958  indstrd  42225  dvgrat  44344  dfatbrafv2b  47275  fnbrafv2b  47278