Theorem sbc8g 3766

Description: This is the closest we can get to df-sbc 3759 if we start from dfsbcq 3760 (see its comments) and dfsbcq2 3761. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbc8g	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))

Proof of Theorem sbc8g

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 3760	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2		eleq1 2898	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
3		df-clab 2799	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
4		equid 2019	. . . 4 ⊢ 𝑦 = 𝑦
5		dfsbcq2 3761	. . . 4 ⊢ (𝑦 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
6	4, 5	ax-mp 5	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
7	3, 6	bitr2i 278	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑})
8	1, 2, 7	vtoclbg 3556	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))

Syntax hints:   → wi 4   ↔ wb 208  [wsb 2069   ∈ wcel 2114  {cab 2798  [wsbc 3758

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  ax-8 2116  ax-9 2124  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-clab 2799  df-cleq 2813  df-clel 2891  df-sbc 3759

This theorem is referenced by:  bnj984  32226  rusbcALT  40856