Theorem sbcie2g 3008

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3009 avoids a disjointness condition on 𝑥 and 𝐴 by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
sbcie2g.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
sbcie2g.2	⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
sbcie2g	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒))

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

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

Proof of Theorem sbcie2g

Step	Hyp	Ref	Expression
1		dfsbcq 2976	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2		sbcie2g.2	. 2 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))
3		sbsbc 2978	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4		nfv 1538	. . . 4 ⊢ Ⅎ𝑥𝜓
5		sbcie2g.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	4, 5	sbie 1801	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
7	3, 6	bitr3i 186	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
8	1, 2, 7	vtoclbg 2810	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1363  [wsb 1772   ∈ wcel 2158  [wsbc 2974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sbc 2975

This theorem is referenced by:  sbcel2gv  3038  csbie2g  3119  brab1  4062