Theorem sbcco3g 4394

Description: Composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2402. Use the weaker sbcco3gw 4389 when possible. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sbcco3g.1	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
sbcco3g	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑))

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

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcco3g

Step	Hyp	Ref	Expression
1		sbcnestg 4392	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
2		elex 3474	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3		nfcvd 2924	. . . 4 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶)
4		sbcco3g.1	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
5	3, 4	csbiegf 3885	. . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
6		dfsbcq 3745	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑))
7	2, 5, 6	3syl 19	. 2 ⊢ (𝐴 ∈ 𝑉 → ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑))
8	1, 7	bitrd 282	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1568   ∈ wcel 2141  Vcvv 3453  [wsbc 3743  ⦋csb 3852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-13 2402  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-v 3455  df-sbc 3744  df-csb 3853

This theorem is referenced by: (None)