Theorem bnj91 32243

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj91.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj91.2	⊢ 𝑍 ∈ V

Assertion

Ref	Expression
bnj91	⊢ ([𝑍 / 𝑦]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑓   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓)   𝐴(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝑍(𝑥,𝑦,𝑓)

Proof of Theorem bnj91

Step	Hyp	Ref	Expression
1		bnj91.1	. . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2	1	sbcbii 3776	. 2 ⊢ ([𝑍 / 𝑦]𝜑 ↔ [𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
3		bnj91.2	. . 3 ⊢ 𝑍 ∈ V
4	3	bnj525 32119	. 2 ⊢ ([𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5	2, 4	bitri 278	1 ⊢ ([𝑍 / 𝑦]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3441  [wsbc 3720  ∅c0 4243  ‘cfv 6324   predc-bnj14 32068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-sbc 3721

This theorem is referenced by:  bnj118  32251  bnj125  32254